/ 
 
Pradical Geometry 
 
 Applied to the Useful ARTS of 
 
 Building,Surveying,Gardening andMenJuration‘, 
 
 Calculated for the Service of 
 
 Gentlemen as well as Artisans, 
 
 And fet to View 
 
 Ill FOUR PARTS. 
 
 CONTAINING, 
 
 Preliminaries or the Foundations pt the feveral ARTS 
 
 above-mentioned, 
 
 II. The various Orders of Architeaure, laid down and improved 
 
 from the beft Mailers j with the Ways of making Draughts of Buildings, Gardens, 
 Groves, Fountains, the laying down of Maps, Cities, Lordihips, Farms, 
 
 HI. The Doarine and Rules of Menfuration of all Kinds, illuK 
 
 trated by feleft Examples in Building, Gardening, Timber, &c. 
 
 IV. Exaa Tables of Menfuration, fhewing, by infpeaion, the 
 
 fuperficial and folid Contents of all Kinds of Bodies, without the Fatigue of Arith- 
 metieal Computation: 
 
 To which is annexed. 
 
 An Account of the Clandeliine Praaice now generally obtainiivc 
 
 in Menfuration, and partieularly the Damage fuftained in felling Timber by Meafure. 
 
 The Whole 
 
 Exemplifi'd with above 6o Folio Copper Plates, by the beft Hands 
 
 B AT T T L A N G L E r. 
 
 LONDON: 
 
 Printed for W. and J. Innys, J. Osborn and T. Longman, B. Lintot, J. Woodman 
 and D.LroNS, C. King, E. Symon, and W- Bell. 1726. 
 
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 ^ 1 - 
 
 V 
 
 
TO THE 
 
 Lord PAISLEY. 
 
 My Lord, 
 
 L L who are acquainted with the Subjeds 
 of the following Treatife will acknow¬ 
 ledge my Judgment in the Choice I have 
 made of your Lordlhip’s Name, which 
 can not fail to recommend it to the peru- 
 fal of the Public ; And though an Author 
 is very unwilling to beleive his Works deftitute of real 
 Merit and Ufefulnefs, yet if this Book lliall meet with 
 Approbation, I am fenfible how much will be owing 
 to your Lordlhip’s Patronage, whofe known Skill in 
 thefe Sciences is the Foundation of this Trouble, 
 
 Permit me to add, that I have a particular Pleafurc 
 in doing myfelf this Honour at a Time when your 
 
 A 2 Lordlhip’s 
 

 D E D 
 
 I C J T 1 0 N. 
 
 Lordflilp’s great Merit has placed you at the Head of a 
 moft Ancient and moft Honourable Society, whofe 
 profound Knowledge, in thefe Affairs, is their Pr.de 
 and Diftinction. I am, 
 
 LORD, 
 
 Tour Lordjhifs moji Obedient, 
 
 Adoji Humble, 
 
 (^nd MoJi De-voted Servant, 
 
 B. Langley. 
 
 r 
 
E F A C E. 
 
 H E fabjefls of the prefent treatife, on account of their antiquity, 
 ufefulnefs and entertaining variety, having been the delight of 
 the greateft mafters in knowledge, thro’ various ages, are, it muft 
 be acknowledged, tranfinitted to us in a fuitablc degree of per- 
 feiftion. They have indeed been largely treated of by various 
 hands, but generally in a theoretical, rather than in a praflical 
 manner, fo as to appear fomewhat intricate and obfcure to fuch as 
 were not acquainted with the principles of mathematics, or have 
 not applied themfelves in earneft thereto. My delign therefore is to treat of archi- 
 tedlure, gardening, menfuration and Land-furveying, in a method as eafy and in¬ 
 telligible as it is new and generally ufeful. I fliall begin with the fundamental, or 
 firll principles of thele feveral arts, and gradually condufl: my reader from the eafier 
 parts of ’em up to the hardeft, taking particular care all along to let him fee the ritile 
 as well as the duke thereof; the fruitful praiSice, and not the barren theory only. 
 From a failure of authors in this point, I apprehend it is that thefe arts are at pre¬ 
 fent much lefs cultivated than they merit. An author cannot do them greater 
 juflice, than to paint them as they arc, maff ufeful and delightful employments ; 
 of great importance in human life. To convince the world of this truth, as it is 
 the defign, fo it wou’d be the higheft recommendation of the prefent treatife. And 
 this I can fincerely fay, that I have had a view thereto thro’ the execution of the 
 whole defign. I fhall not therefore offer at any recommendation of the arts them¬ 
 felves, which want no able hand to fet them off with colours, and the winning 
 charms of rhetorick but leave my reader, from the plain, naked, artlefs fads and 
 obfervations he will meet with in the work, to determine of their merit. And I 
 am greatly miftaken if to all true judges this does not appear a more equitable, and 
 more unexceptionable procedure than to write, as the ufual manner is, an encomium 
 of the arts I treat of, in order to recommend the work. For if the book cannot be 
 fupported by its own merit, I am fure a panegyrick upon its fubjeft will but render 
 it the more ridiculous and contemptible. All that I requell is a fair and candid pe- 
 rufal. I defire only that my reader wou’d come with a mind prepared not to be 
 flartled, or prejudiced againlt the author, by the appearance of novelty back’d with 
 reafontho’ it at firft fight fhou’d feem to thwart fome current and prevailing opi¬ 
 nions. This were a temper that wou’d for ever exclude the light, and dronilhly re- 
 
 a main 
 
a PREFACE. 
 
 main content with whatever dodrine happens to have its run. But we of late have 
 feen luch fuccefsful inrodes made into opinions once thought juft, that we cannot 
 be too fufpicious of our entertaining eftablifh’d errors for truth, and fhutting our 
 eyes againft plain faft and obvious reafon. 'Tis not that I pretend to a faculty 
 beyond that of others in difcovering the truth in the particulnr fubjefls I have here 
 treated j but my genius leading me to fuch kind of ftudies, I hope I may be allowed 
 to have obferved the common things, and to make my own ufe of them. If what I 
 alledge be true, (for which 1 always give my reafons) the world will have the ad¬ 
 vantage ^ but if it fhall prove to be falfe, I lhall willingly bear the blame: Onlv 
 I make this requeft, that I may be cenfured by the proper judges, and fuch as have 
 been converfant in the fame kind of ftudies with my felf • otherwife the world I 
 hope, will agree with me, that I am condemn’d unjuftly. That the reader may form 
 the better judgment of the performance, he may be pleafed to take the following 
 account thereof. ° 
 
 Geometry being the balls of architeefture, gardening, menfuration and land furvey- 
 ing, (which are the fubjefts of this treatife) I have in the firft part, laid down all the 
 molt ufeful and neceftary geometrical definitions, problems, theorems, and axioms, 
 that are abfolutely neceifary to be well underftood by every one who defires to be 
 a complete artifan, and thofe in a moft concife and familiar manner. The fecond 
 part contains the application of the firft to praclife in the geometrical conftruftion 
 of all kind of fcales for the delineating, and menfuration of all forts of plans and 
 uprights, and of the Ttifian, Dorick, lonick, Corinthian, Cnmpofite, Freticb and 
 Spanijh Orders of architedlure, with their derivation, proportion, (^c. in general 
 And feeing that neither ancient or modern architefl.s have yet agreed on the meal 
 fares of the principal parts of entire columns: I lhall therefore before I proceed 
 any further, demonftrate the fame particularly. 
 
 The principal parts of entire columns are three, vis. The pedeftal, the column 
 and the entablature ; all which are feverally divided into three other parts. As 
 firft, the pedeftal by its bafe, die, and cornice ; the column by its bafe, lhaft and 
 capital, and the entablature by its architrave. Freeze and cornilh, whofe feveral 
 heights and projedlures are meafured by modules and minutes. ( Fide prob. the oth 
 feci. I. part ad.) 
 
 ([.) Pedeftals, (called by the ancients Styloiats) are of two kinds, viz. The one 
 broken, and the other continued. Broken pedeftals, are parts of a continued pe¬ 
 deftal, which projeft or break out, right under each column, as in the Theatre of 
 Jlfei celhts, the arches of Titus, Septiwius^ and Conjinntine in the ColiJ'emtt. and in 
 the altars of the Pmtheon. Continuea pedeftals are luch as range throughout with¬ 
 out projedlures or breaks under each column, as in the Goldfmith's arch, the’temple 
 of Vejla at Tivoli, and that of Fortuna Virilis. ’ ‘ 
 
 Both ancient and modern architedls have delivered rules for the heights of entire 
 pedeftals, but all different, whereby the young architedl is at a lofs to know a- 
 mong the feveral, which is the beft. ’ 
 
 Tiilladio makes the height of the Tufean pedeftal three modules- the Dorick four 
 modules and five minutes; the. lonick five modules four minutes ^ the Corinthian fiyo 
 modules one minute, and the Compofite fix modules feven minutes. 
 
 Scamnwz&i makes the height of the Tufean pedeftal three modules twelve mi¬ 
 nutes ^ the Dorick foot modules eight minutes^ the lonick five modules- the Co¬ 
 rinthian fix modules eleven minutes, and the Compofite, fix modules two mi¬ 
 nutes. 
 
 Vignola 
 
PREFACE, 
 
 iii 
 
 Vignoh makes the height of the Tnfcan pedeflal five modules ; the Dorick five 
 modules four minutes j the lonick fix modules, and the Corinthian and Compojite 
 feven modules each. 
 
 Serlio makes the height of the Titfcan pedeftal four modules fifteen minutes; the 
 ‘Dorick fix modules the lonick fix modules ^ the Cormthian fix modules fifteen mi¬ 
 nutes, 3 nd the Compofite feven modules four minutes. The height of the lonick 
 pedeftals at the temple of forttma Firilis, is feven modules twelve minutes j thofe 
 of the theatre of Marcellus three modules eight minutes, and at the Colifewn four 
 modules twenty two minutes. 
 
 The height of the Corinthian pedeftals, at the altars of the Pantheon, are feven 
 modules twenty eight minutes the Colifetitn four modules two minutes, and the 
 Compofite pedeftals of the Goldfiniths-arch, feven modules eight minutes. 
 
 Now fince ’tis abfolutely neceflary, that thefe diverfities ftiould be reduced to a 
 mean proportion, for a ftandard meafure, therefore I have done it, and is as fol¬ 
 lowing, vis,. Make the entire height of the Titfcan pedeftal equal to two diame¬ 
 ters or modules. ^ The Dorick to two modules twenty minutes j the lonick to two 
 modules forty minutes ^ the Corinthian to three modules, and the Compofite to 
 three modules twenty minutes, the progreflion being of forty minutes. 
 
 N. P. Tha t a module is a length equal to the diameter of the bafe of the column ' 
 divided into lixty equal parts called minutes. ’ 
 
 The difference in the proportions of the parts of pedeftals, are as great as thofe 
 of their heights, and therefore I have alfo eftabliflied this one general proportion 
 for the parts of all pedeftals, vis. Divide the height of any pedeftal ( be it Tttf- 
 can, Dorick, lonick, Corinthian or Compofite^ into one hundred and twenty equal 
 parts, and of thofe parts give to the focle twenty, to the mouldings of the bafe 
 ten j to the die or trunk feventy five, and to the cornice fifteen. 
 
 The proportions afligned for the projeefture of the bafe and cornifli of pedeftals, 
 by both ancient and modern architeds, are as various as their other parts, and 
 therefore in this point alfo, 1 have reduced the diverfities to a mean proportion, 
 that thereby a general rule may be obferved throughout the five orders, vis. In 
 any order, be it Tnfcan, Dorick, lonick, Corinthian, or Compofite, make the bafes 
 of pedeftals, (excluiive of their zocolo or plinth) with a projefture equal to their 
 altitude, which being different in every order, will thcicfore caufe the projedure 
 of the bafe to be different in each order. In the projedure of cornices of pedef¬ 
 tals, made by either ancients or moderns, there is but little difference, for^ they 
 ufually make their projedure equal to (or very little more, than) that rf the bafe 
 of which the laft is to be preferr’dand therefore, for a ftandard rule, I give the 
 following proportions, for the projedure of both bafe and cornice, as follows vis. 
 To the Dorick pedeftal, I give to the projedure of the bafe twelv'e minutes, and to 
 the cornice fourteen. To the lonick pedeftal I give to the projedure of the bafe 
 fourteen minutes, and to the cornice feventeen. To the Corinthian pedeftal, I give 
 to the projedure of the bafe fifteen minutes, and to the cornice nineteen. And 
 laftly, to the Compofite pedeftal, I give to the projedure of the bafe fixteen mi¬ 
 nutes, and to the cornice twenty two. 
 
 I fhall now mention another particular belonging to pedeftals, as is common in 
 all the orders, (and then proceed to the proportions belonging to columns) which is 
 as follows : 1 hat the breadth of the die of every pedeftal be always equal to the 
 projedure of the bafe of its column. 
 
 The 
 
PREFACE. 
 
 tv 
 
 The projefture of the bafes of columns, is alfo an unfettled part of architecture. 
 For to the litfcan bafe Palladio and Scamtnozzi allow each forty minutes, as alfo 
 hath the Trajan’s column, but Vignola allows forty one, and Serlio forty two mi¬ 
 nutes. To the projefture of the ‘Dorick bafe, Palladio allows forty minutes, 
 Scawmoazi forty two, Vignola forty one, Serlio forty four, and at the Colifeimi for¬ 
 ty minutes. 
 
 To the projeclure of the lonick bafe, Palladio, Scamnwazi, and Serlio allow 
 forty one minutes ^ Vignola forty two the temple of Manly tortnne forty three ■, 
 and the. Colifemn forty minutes. To the projeclure of the Corinthian bafe, Scammoz- 
 zi, Serlio, and the Colifeum allow forty minutesthe portico of the Tantheon forty 
 one j the three columns oi Campo Vaccine, the baths of Dioelefian, Palladio, and 
 Vignola allow each forty two, and the pilafters of the portico of the Pantheon for¬ 
 ty three. 
 
 And laftly, to the projeflure of the Compofite bafe, the temple of Bacchus, 
 the arch of Septimius, Scammozzi, and Serlio, allow each forty one minutes, 
 Palladio and Vignola forty two ; the baths of BiocleJIan forty three, and the 
 arch of Titus forty four minBtes. Hence it appears, that forty two minutes is a 
 mean proportion, between the extremes, and is what I recommend for the pro- 
 jefture of the bafes of columns in general. 
 
 The great diverfity of the lengths of columns of the fame order affign’d by ar- 
 chitecls, is a very difficult point to account for : To the length of the Tufean 
 column Vitruvius Palladio, and Vignola, give feven diameters or modules. Scammoz- 
 zi feven and an half. The Trajans column eight, and Serlio but fix diameters. 
 
 To the length of the Dorick column, Vitruvius in temples, allowed feven dia¬ 
 meters ; but in Portico’s of temples feven diameters and an half ^at the Colifeum they 
 confifl: of nine diameters and an half j and at the theatre of Marcellus feven modules 
 and fifty minutes ; Scammozzi gives eight and an half ■, and Vignola eight diameters 
 only. 
 
 To the length of the lonick column, Vitruvius, Palladio, and Serlio, give 
 eight diameters forty minutes, as alfo is the Colifeum^ and theatre of Marcellus 
 at Rome. 
 
 To the length of the Ccri/ithian column, Vitrwvitis gives nine diameters thirty 
 minutes, and Serlio nine Jiameters only ; at the porch of the Pantheon they are 
 nine diameters thirty fix minutes. 
 
 The temple of Pantheon they are nine diameters, thirty fix minutes. The 
 temple of Vejla nine modules thirty nine minutes. The temple of the Sibil eight 
 modules fixteen minutes. The Colifeum eight modules thirty feven minutes. 
 
 The temple of Peace nine modules thirty two minutes : The arch of Conflan- 
 tine eight modules thirty feven minutes; The three columns of the Campo Vac¬ 
 cina, ten modules fix minutes j the porch of Septimius nine modules thirty eight 
 minutes ^ the temple of Faujlina nine modules thirty minutes, and the Eaflic of 
 Antonhius ten modules exaftly. 
 
 To the length of the Compofite columns, Scammozzi gives nine modules forty 
 minutes, and the fame is at the temple of Bacchus ; the arch of Septimius nine 
 modules thirty minutes, and the arch of Titus ten modules precifely. 
 
 Now 
 
PREFACE. 
 
 V 
 
 Now, becaufe ’tis reafonable, that a proportionable length fliould be eftablilhed 
 for the’length of columns in general, I have therefore reduced the extremes of 
 their diverfities to a mean proportion as following, viz. Make the height of 
 the Tufcan column equal to feven diameters or modules, and twenty minutes; 
 the Dorick to eight modules; the lonick to eight modules forty minutes; the Co¬ 
 rinthian to nine modules twenty minutes, and the Compofite equal to ten modules 
 only. So will their progreflion be proportionable, confifting of forty nainutes in 
 each column. 
 
 The diminilhing of columns being firfl aflign’d for that beautiful appearance, 
 as flows therefrom, is made in three different manners. As firff, to begin the 
 diminution at the bafe of the column, and continue it to the capital. Ihe fe- 
 cond is to make the column thicker towards the middle, than at its bafe, dimi- 
 nifhing of it towards the bafe and capital, which kind of diminution is called the 
 fwelling. The third and laft way is according to the antique manner : Beginning 
 the diminution at one third of the height above the bafe of th® column, as is 
 Ihewn in folio 63 hereof, and is the molt beautiful kind of diminution. 
 
 The difference and quantity of diminution in each of the orders, is exhibited 
 in fedl. 2. Part 2. hereof. 
 
 There being yet no certain determin’d proportion for the height of the aflra- 
 gal and cincture, which terminate the fhaft of a column. I therefore have alfo 
 reduced thofe members, to fuch a proportion, as may be applied throughout the 
 orders in general. As firfl to the cindlure, I give three minutes, and to the aftra- 
 gal three minutes and one third. In the Pantheon, the temples of Vefta and Ma 7 i- 
 ly Fortune, and arch of Titus, the cindlures ate very near three minutes. And in 
 the temple of Antonhiiis and Fauftina fomething more, as alfo the temple of 
 Bacchus, the arch of Septiinius, and in the bath of Dioclefian, from which I have 
 extrafled my mean proportions. 
 
 The received proportion for the height of the bafes of each order is, to make 
 them equal to the feraidiameter of the column at its bafe, the Tufcan excepted, 
 in which the cinfture is included, which in other orders is not. 
 
 In the five orders of architeflure, there are three different heights of Capitals. 
 The Tufcan and Dorick capitals are always equal to the height of their bafe. 
 The lojiick capital, from the top of the abacus to the point of interfedion, where 
 the cathetus and voluta interfed each other, at the bottom of the volute ; to the 
 femidiameter and an eighteenth part thereof And laflly, the Corinthian and Com- 
 poftte capitals to one module and ten minutes. 
 
 But, notwithftanding that thefe meafures are affigned for the height of capi- 
 
 als, yet ’tis to be obferved, that the ancients did not obferve them ftridly. 
 For the capital of Trajan's column (which is of the Tufcan order) is lefs than 
 the femidiameter of the columns bafe, by a full third ; and in the Lorick capi. 
 tal of the theatre of Marcellus, its height is almoft thirty three minutes, and 
 that of the Colifeum near thirty eight minutes: Nay, in the Corinthian cafml 
 oi Vitruvius (the father of ardiiteds) he makes its height but fifty minutes; 
 And therefore I finding that its height was not fufficient, have introduced a mo¬ 
 dern capital in its Head. At the temple of the Sibyl at Tivoli, the height of the 
 Corinthian capitals are but forty feven minutes. In the frontifpicce of Nero (ixty 
 fix minutes, and in the temple of Vefta at Rome almoft fixty eight minutes. 
 And laftly, the height of the Compofite capitals of the arches of Septimius and 
 the Goldfmiths, are but fifty eight minutes and an half, and the temple of 
 
 b Bacchus 
 

 vi PREFACE. 
 
 Bacchus fixty fix. Hence appears the oppofite diverfities, from which are efta- 
 bliih’d the mean proportions before delivered. 
 
 And altho’ the proportions of the aforefaid capitals are very different from 
 each other, yet there is a far greater difference in the height and projeffure of 
 entablatures, which they with their columns fupport. ‘ 
 
 To the height of the Tufcan entablature, Vitruvius allows one hundred and five 
 minutes, Palladio one hundred and four minutes, Scamozzi one hundred and 
 twelve, Vignola one hundred and five, and Serlio ninety minutes. 
 
 To the height of the Borick entablature, Vitruvius allows one hundred and twen¬ 
 ty minutes ( equal to two modules ) Palladio one hundred and thirteen minutes: 
 Scamozzi one hundred and twenty feven minutes ■, Vignola one hundred and 
 twenty ; Serlio one hundred and twelve j and the like of all other Mafters as are 
 fet forth in the 19, 21, 22, 24, 25, 26, 27, 28, and 29th plates hereof, to which 
 I refer. 
 
 Now feeing that the beauty of an order doth confift in a proportionable en¬ 
 tablature ; therefore to prevent the deftruftion thereof, by having entablatures ei¬ 
 ther of fuch a fize: that they feem utterly infupportable, as thole of Campo Vac- 
 cino, and the frontifpiece of Kero, or on the contrary, too mean and pitiful as 
 the entablatures of Bullant and Belorme ; I advife that the height of all en¬ 
 tablatures be always equal to two diameters, or one hundred and twenty mi¬ 
 nutes, and their projeflure of the cornice, equal to the height thereof in the 
 Tufcayi, lonick, Corinthian and Compojite entablatures, and the Tiorick entabla¬ 
 ture alfo, when the cornice is made without rautules, (as in that famous ftruc- 
 ture the CoUfeum ). But when the ‘Dorick entablature hath mutules introduced 
 their length requires the entire cornice to have more projeffure than height. ’ 
 
 Having thus demonffrated the proportions of the principal parts of columns 
 I fhall now proceed to the remaining part of my preface. ’ 
 
 The third feclion of part 2. contains many excellent architeftonical axioms and 
 inalogies, colleiffed from molt grand mafters. 
 
 The fourth feftion of part 2. contains the ufe of an infpeflional plain fcale 
 which furnilhes the young ftudent not only with all kind of fcales, but readily 
 divides the feveral parts of a building inftantly. ^ 
 
 The fifth feftion of part 2. contains trigonometrical definitions, with the con- 
 ftruftion of chords, fines, tangents, half tangents, fecants and verfed fines ap¬ 
 plied to practice in the folution of the twelve cafes of plain trigonometry which 
 is performed geometrically alfo, by the help of a plain fcale and pair of com- 
 palTcs, in a very concife and familiar manner. 
 
 The fixth fedion of part 2. contains the geometrical conftrudion of draughts 
 plans, maps of gardens, farms, d'-c. Wherein is fliewn how to perform fuch 
 works, much more expeditious and exafl than any author yet extant. 
 
 The third part contains all the moft ufeful geometrical axioms and analogies 
 for the raenfuration of any fuperficial figure or folid body. ^ 
 
 The fourth feftion of part 3. contains the meafures, and manner of taking the 
 dimenfions of all kinds of work relating to building, as Carpenters Glaziers 
 Joiners, Painters, Plafterers, Mafons, Bricklayers, Paviors, &c. ’ ’ 
 
 The 
 
PREFACE. 
 
 vii 
 
 The fourth part contains divers infpeftion tables of menfuration, whereby 
 any diinenlion may inftantly be call up, without the alfiftance of multiplication, 
 or even fuch capacities as are not mailers of crofs multiplication, are hereby 
 enabled to meafure any work with as great accuracy, as the belt accomptant. 
 
 The firft, fecond, third, fourth, fifth, fixth, feventh, eighth, ninth, tenth, ele¬ 
 venth, twelfth, fifteenth, fixteenth and feventeenth plates, being thofe which the 
 feveral fubjeds hereof have recourfe to, need not in this place fay any thing 
 thereof. 
 
 The thirteenth and fourteenth plates contain a new fyftem of gardening, 
 wherein ’tis Ihewn what great improvements may be made, even in the 
 fmalleft of gardens ^ for by the method there obferved, a fmall garden may be 
 made to appear as a very large one ; and fuch as are very large, to become the 
 moll noble and delightful. 
 
 And becaufe no gardener can well underfland the true manner of laying out 
 a garden, even in any manner as bears any proportion, without well underftand- 
 ing the elements of geometry •, therefore for his fake, in the firft part hereof, 
 1 have laid down all as is neceffary to be known in a moft concife and eafy 
 manner, and applied to pradice in the geometrical conftrudion of all kind of 
 lines and figures, as are requifite for his purpofe in the pradice of gardening. 
 Perhaps that fome may exped that I ftould herein treat of the culture of 
 lands, the management of fruit trees, &c. which are parts as doth not re¬ 
 late to the mathematical part of gardening, as in defigning, drawing, laying out, 
 C^c. But if God permits, I lhall fpeedily communicate a treatife thereof, where¬ 
 in I lhall difeover many curious experiments, as will prove both pleafant and 
 advantagious to all lovers of gardening. 
 
 The eighteenth, nineteenth, twentieth, twenty firft, twenty fecond, twenty 
 third, twenty fourth, twenty fifth, twenty fixth, twenty feventh, twenty eighth 
 and twenty ninth plates, contain the geometrical profiles and elevations of the 
 five orders of architedure, as laid down by all the grand mailers, both ancient, 
 antique, and modern. 
 
 The thirty and thirty firft plates are defigns for the entrance into lhady 
 walks j the firft into a right lined walk, and the other into a curved, or arti- 
 natural walk, and thofe are delineated according to the truth of perfpedive. 
 
 The thirty third plate contains two defigns for the enterances into Grottos 
 according to the grand manner. 
 
 The thirty fourth plate contains divers capitals of the Corhithian and Co7ii- 
 fofite orders, taken from the works of Vitruvius, and Andrew Boffe, with an 
 elegant elevation of a noble ftrudure after Palladio. 
 
 The thirty fifth, thirty fixth, thirty feventh, and thirty eighth plates con¬ 
 tain divers geometrical elevations of doors, neathes, Oc. of the Jufean, Do- 
 rick, lonick, Corinthian and Compofite orders, adorned with His moft Sacred 
 Majefty King GEORGE ; their Royal Highneffes the Prince and Princefs, 
 whom God preferve. 
 
 The thirty ninth plate contains divers excellent defigns for chimney-pieces, 
 collefled from the bell of mailers. 
 
 And 
 
PREFACE. 
 
 viij 
 
 And the fortieth plate, the geometrical elevation of the portico of St. Mary 
 the Egyptian, with a Corinthian frontifpiece from the Ancients, and the impofts of 
 the Tufcan, ‘Dorick, lonick, Corinthian and Compoftte orders. 
 
 Having thus, by way of preface, explain’d the feveral parts of the work, I 
 now recommend you to the praftice, defiring that you wou’d read and examine 
 it, without critical envy, free from pre-occupation that may obfcure your Judg¬ 
 ment, and hinder your acknowledging the truth of what I have here prefented 
 for your improvement. 
 
 Therefore be not advifed by fijch as condemn a conception when they under- 
 fland it not ^ and believe it falfe becaufe ’tis new; neither imitate thofe, who 
 feeking only to carp at words, negleft the fenfe of the fubjeft. 
 
 B. Langley. 
 
 THE 
 
OF 1' H E 
 
 CO N T E N T S- 
 
 part I« 
 
 SECT. I. 
 
 Of Geometrical D E F T. is! 1 T I O N S. 
 
 Definition 
 
 I a. Toint 
 
 j O Of a Line 
 
 3 Of Lines 
 
 ^ Of a Right Line 
 
 i Of n Circular Line 
 
 <5 Of an Elliptical Line 
 
 ■j Of a Parabolical and Hyperbolical Line 
 
 8 Of the Ttrmmation of Lines and Superficies 
 
 9 Of Superficial Figures 
 10 Of a Circle 
 
 II Of the Center of a Circle 
 
 12 Ofthe ‘Diameters of a Circle 
 
 Page 
 
 I 
 
 15 Of 
 
 * 
 
The C O N T E N T S, 
 
 Definition 13 Of the Radius of a Circle_ 1 
 
 14 Of the Sellion of a Circle j 
 
 15 Of a Semicircle | 
 
 16 Of a c^iadrant ' 
 
 ty Of the Radius of a fhiadrant 
 
 IS OfanEUipJis 
 
 19 Of the ‘Diameter of an EUipfis j 
 
 20 Of a Triangle 1 
 
 zi Of a Geometrical Square, Ritombus, Rhom- 
 
 boyades, dWi/Trapezium j 
 
 2} Of Irregular Figures -j 
 
 23 Of a Pentagon, Hexagon, Heptagon, Octagon, i!, 
 Nonagon and Decagon 
 
 24 Of theDizgoml Lines of a Geometrical Square 
 z-i Of the Diameters of a Geometrical Square 
 
 26 Of Center of a Geometrical Square 
 
 27 Of the Termination of Solids 
 
 2 8 Of a Solid Body ' > 
 
 29 O/GftfWf/rrW Solid Bodies 
 
 30 Of a Sphere 
 
 31 O/flConc 
 
 32 Of the Fruftrum of a Cone 
 
 3 3 Of a Pyramis, or Pyrament 
 
 34 O/aPrifm 
 
 35 (y a Tetraedron , ’ 
 
 36 Of a Cube ' 
 
 37 Of Fruftrum ofaCnht 
 
 38 Parallepipedon 
 
 39 Of an Oftaedron 
 
 40 Of a Dodeeaedron s 
 
 41 0 /Icofaedron 
 
 42 Of the Bafts of a Sphere 
 rr'i Of the Bafts of a Cylinder 
 44 Of the Bafts of a Cone 
 
 s E C T. I/. 
 
 Of Geometrical P R O B L E xM S, 
 
 Problem 1 divide a right Line into two equal Parts by 
 
 -*• a Perpendicular 
 
 2 Upon any ‘Point in aright Line giveuy to ereU a 
 Perpendicular 
 
 3 From the End of agivenUne, toereSl a Perpen¬ 
 
 dicular ‘ 
 
 4 To perform the preceding another JFay 
 
 5 To let fall a Perpendicular/w» a Point^rwj 
 
 6 To deferibe Parallel Lines 
 
 7 To make an Angle equal to an Angle given 
 
 s 
 
 9 
 
 i To 
 
The CONTENTS. 
 
 Problem 
 
 8 To divide an Angle into tivo equal Tarts 
 
 9 To divide a Right Line into any .Number of equalt 
 
 Tarts '■ 
 
 10 To find a Mean Proportion between two Rioht( 
 
 Lines given ° 
 
 11 To defcribe a Circle as fimll pafs through three 
 given Points 
 
 I a To inficribe a Triangle Geometrical Square, Penta¬ 
 gon, Hexagon, Heptagon, Oflagon, nomgon, andi 
 Decagon within a Circle ' 
 
 To make an Equilateral Triangle 
 
 14 To make a right lined Triangle equal to threeL 
 
 given right Lines C 
 
 15 To defcribe a Geometrical Square 
 
 16 To defcribe an Oblong S 
 
 17 To defcribe a Rhombus \ 
 
 18 To defcribe a'MsovcSoo'jzdiC.i r 
 
 19 To defcribe a Trvpe.7\\xm y 
 
 ■2.0 To defcribe an Ellipfis r 
 
 21 To defcribe any Ellipfis 
 
 22 ToinfcribeaChdcwithin a S<i\i2,rc > 
 
 23 To infcribe a Circle within a Circle, & contra 
 
 24 To infcribe a Circle within a Triangle 1 
 
 25 To circimfcribe a Circle Triangle I 
 
 26 Two Points within a Cirdzgiven, to defcribe an- ^ 
 other Circle, as jhall divide the Gircumference of 
 the given Circle into two equal Tarts 
 
 27 To make a Geometrical Square equal to any Trian-^ 
 
 ^e.given ) 
 
 ■2.% To make a Geometrical Square equal to any Paral- ( 
 Idlogram \ 
 
 29 "n divide aright Une. in any Proportion required') 
 
 30 To divide the Circumference of any Circle into> 
 
 3 60 equal Tarts (or Degrees) ) 
 
 II To infcribe an'EWi^fiswithin an Oblong ) 
 
 3 2 To erebl a Perpendicular, by the Help of a ten Foot> 
 Rod V 
 
 j I 
 
 18 
 
 19 
 
 21 
 
 SECT. III. 
 
 Of Geometrical Axioms and Theorems, from Folio 
 to Folio 
 
 S E C T. IV. 
 
 Of the Coiiftruftion of Compound Geotneirtccil 
 figure S. 
 
 /tXioms 1 , II, and III. 
 
 Axioms IV, V, VI, VII, VIII, IX ^6 
 
 A general Rule concerningCompound Figures *7 
 
 Eight Problems to dejcribe compound Figures, from Folio 
 27, M 30 
 
 SECT. 
 
The CONTE N T S, 
 SECT. V. 
 
 Of the ConftrucTion of Compound Lines. 
 
 1 7''0 deferibe a fmgle Spiral Line 
 
 2 To deferibe a double S’^iiillAnc 
 
 page 
 
 t 31 
 
 S 
 
 3 To deferibe the running Worm 
 
 3i 
 
 4 To deferibe a treble Spiral Line 
 
 3J 
 
 5 To deferibe a quadruple Spiral Line 
 
 35 
 
 6 'To deferibe an elliptieal Spiral Line 
 
 35 
 
 ■j To deferibe a Scrole 
 
 3/ 
 
 %To deferibe an Artinatural Line 
 
 3S 
 
 SECT. VI. 
 
 
 40 
 
 41 
 
 Of the Geometrical Contruncation of the Cube, and the 
 Solids generated thereby. 
 
 T O divide any right Line in extream and mean Pro-... 
 fortiori S 
 
 Cube ) 
 
 Canted Cube 
 Fruftum of a Cube 
 Tetraedron 
 
 Fruftum of a Tetraedron 
 Octaedron 
 
 Dodecaedron f 
 
 Icofaedron C 
 
 11 Rhombs J 
 
 13 Rhombs 
 
 P A R T II. 
 
 SECT. I. 
 
 Of the Geometrical Conftruaion of Tlans and Uprights. 
 
 p robl em I 7^0 make divers Scales of equal Parts 
 
 1 ^ To make any Line (or Scale) of Chords 
 
 3 To make aV\ia. equal to a Plin given 
 
 4 d fccond Example 
 
 5 To meafure the §uantity of an Angle, by the Help 
 of a fwo Boot, five Foot, See. Rod only 
 
 46 
 
 47 
 
 49 
 
 50 
 
 51 
 
 6 To 
 
Ihc C O N T E N T S. 
 
 Probl EM 6 7 b take the Plan of any crooked Line, tvhichis not p 
 any part of an Ellipfis or Circle C 
 
 7 Ho'^ to take the Plan of any Building 
 
 8 Howtodra-ju the Geometrical Uprgiht (or Front) p 
 
 of any Building ^ f 
 
 9 'To delineate the Geometrical Upright of any of the ■) 
 
 five Orders of Architecture > 
 
 10 To delineate the Tx'ifiy fixes of theXDonck. Order ^ 
 
 11 'To deferibe the Uptight and Inverted Cima p 
 
 iz To delineate Geometrical Upright of anyYx'xxK- > 
 
 ter or Column, with its Entablature ) 
 
 To delineate //t'c Geometrical Upright of any wreath’d 1 
 Waved or twilled Column j 
 
 14 HoitJ to divide the Breadth of any Pilallcr into its ) 
 
 Flutes and Fillets, and to delineate the Geometrical r 
 Upright of the fame ) 
 
 15 To divide the Bafis or Plan of the Shaft of a Column 1 
 
 into its 24 Flutes awrf 24 Fillets f 
 
 16 'To per form the fame voithout Fillets, as is ufual in I 
 
 the Doriefc Order r 
 
 17 To deferibe ow Paper-drawing. Wall, ific. Geo¬ 
 metrical Upright Column, vj'ithitsand[ 
 Fillets 
 
 IS 7 b deferibe the like without Fillets 
 
 19 7 b perform the like, according to Vitruvius 
 
 20 To perform the like according to Vignola 
 
 21 To divide the Bafc of the Shaft of a Column into 
 its cabled Flutings 
 
 22 To div'ide the Bafe of the Shaft of a Column into ' 
 
 Its 24 Flutes, and 24 Fillets, after the Manner oft 
 the Columns withinthe Pantheon * 
 
 23 To deferibe the lonick Voluta, according to the An¬ 
 tique Manner 
 
 Page 
 
 51 
 
 52 
 55 
 
 5 S 
 
 61 
 
 66 
 
 68 
 
 69 
 
 70 
 
 71 
 
 72 
 
 SEC 1. II. 
 
 Ot the DerivatiolU Proportion, Diminution, of the 
 five Orders of Architeflure ixom Folio 73, to 77, 
 
 SECT. III. 
 
 Of Architedlonical Axioms and Analogies. 
 
 Doors 
 
 Of Windows 
 
 Of Gates 
 
 Of Halls 
 
 
 If 77 
 
 Of Galleries 
 
 Of Antichambers 
 
 
 > 7* 
 
 z 
 
 B 
 
 Of 
 
The CONTENTS. 
 
 
 Page 
 
 Of Chambers 
 
 Of Floors 
 
 Of Hall Chimneys 
 
 > 79 
 
 Of Chamber Chimneys 
 
 Of Chimneys m Studies 
 
 
 Of the Funnels of Chimneys 
 
 Of Joifts, Rafters and Girders 
 
 C So 
 
 OTbrair-Cafes 
 of Materials 
 
 8 I 
 
 SECT. IV. 
 
 Of the Defeription and Ufe of [an InfpeclionalTlain 
 Scale, from Folio 8a, to Folio 8d. 
 
 SECT. V. 
 
 Of Tlain Trigonometr/, from Folio 8(J, to 97. 
 
 SECT. VI. 
 
 Of the Geometrical Conftrnliion of Draughts, Tlans^ 
 Maps of Lands, Gardens, Farms, Buildings, &c. 
 
 P R o B L E M I 'T’'o ‘I “V '^'7 Lietd 9g 
 
 2 To makeaVX^a of any Garden, tGddernefs, 5 s.c. loo 
 
 3 To make the Map of any Eftatc, Farm, Lordfhip, cFr. 102 
 
 4 How to increafe, or decreafe anyUraught atpleafnre 103 
 
 5 How to defcribe (yine^ account for) the ’Diminution 
 of the Breadths of Long Walks, Avenues, ire. 
 
 6 To defcribe {and account for) the Diminution of ? 
 
 Objefts in a Landskip 3 
 
 7 To proportion ’i'ci.iysiy, on any'i.ddxz.z ic6 
 
 PART III, 
 
 SECT. I. 
 
 Of Crofs Multiplication 
 
 107 
 SEC T. 
 
The CONTENTS. 
 
 SECT. II. 
 
 Of Geometrical Axioms^ for the Menfuration of Lines 
 and Superficial Figures. 
 
 r R o B LE M I 'L’O meafure a Gcoinetrical Square 
 
 2 To meafnre a Parallelogram 
 
 3 To meafure a Triangle 
 
 4 To (2 Trapezium 
 
 5 To meafure any irregular Figure 
 
 6 To meafure any regular Polygon, as the Pentagon, I 
 
 Hegon, Heptagon, fi'C. t ^ 
 
 7 The Side of a Pentagon, &c. given, to find the Sz- f 
 
 rnidiameter of a Circle infcribed therein I 
 
 . 8 To meafure any C'vlcXc, or any of its Parts, drr. I 
 
 9. To meafure atiy^lW^iii • 'y 
 
 10. To meafure the Superficies of any Sphere, or He- > iit 
 
 mifphcre 
 
 11. do me afire the fupeificial Content of any 'I. 
 Cone 
 
 12. To meafure the ia'pztiieizX Consent of any Pyramis | 
 
 13. Ttf meafure fupcrfidal Content of any Cylm- > 114 
 der 
 
 14. To meafure the fuperficial Content of any Frag¬ 
 ment, or Part of a Globe, or Sphere 
 
 SECT. III. 
 
 Of Geometrical Axioms for the Menfuration of 
 Solid Bodies. 
 
 I ' / 'O meafure the Solidity of a Cube ~i 
 
 3 -* To meaf&e the Solidity of any Vy 
 
 , j - yramis or Cone ( 
 To tneafire the Solidity of the Fruftum of any Pyra- y 
 mis or Cone ' C 
 
 To meafure the Solidity of any Sphere, Globe frc. \ 
 Tiiif Solidity of a Sfaete being given, to find its 
 meter or Axis 
 
 A Segment, or Potion ofa ^fnete being given, to find^ 
 Axis ■ 
 
 To meafure the Solidity Cylinder 
 To meafure the Solidity of any Prifiii 
 To meafure the Solidity Of any Mount 
 Canal, fyc. 
 
 I i: 
 
 Terrace-walk, C Hi 
 
 SECT 
 

 The CONTENTS. 
 
 SECT. IV. 
 
 01 the feveral Meafures and Manner of taking the Di- 
 inenhons of 
 
 ^^Arpenters Work 
 
 Glaziers 
 
 Page 
 
 'J 
 
 > 119 
 
 Joiners Work 
 
 J 
 
 Painters Work 
 
 > 120 
 
 Plaiftcrers JFork 
 
 3 
 
 Mafons Work 
 
 
 Bricklayers Work 
 
 122 
 
 SECT. V. 
 
 Of the xManner of calling up the Dimenfions of Land. 
 
 Problem i j^Eafure taken ij Gurtitt s Chain 12 . 
 
 2 The Vlan of a Piece of Land, with the'\ 
 
 Areagfven, to find the Scale bycohich ‘‘twasplan’dL 
 fippojmg 'tit-as loft ‘ t 
 
 4 Of the Menjuration ofTiirfjor Gardens, &c, J 
 
 SECT. VI. 
 
 Of divers Analogies or Troportions in Land-ineafure, from 
 Folio 12d, to 127. 
 
 part IV. 
 
 Of Infpeclional Tables of Menjuration, from Foljo i 
 
 to jjd. 
 
 28 
 
 Folio '°9 Line 4 . readPto,,6,forHiiK 7, ‘ 
 
 115 Lmei. rad P/«fe 16. for Phte 17. 
 
 ,ri). Problem!, read Pf«» la. for Phte it. 
 
T H E 
 
 PRACTICE 
 
 O F 
 
 Arc hit eel lire, Gardening, Men fur at ion, and 
 
 Land-Surveyings Geometrically demonftrated. 
 
 PART I. 
 
 Of fuch Geometrical Elements as are abfolutely 
 necelTary to be well iinderftood by every Per- 
 fon who delires to well underftand the T ruth 
 of Lineal Architecture, Gardening, and 
 Mensuration univerfally. 
 
 E c T. 
 
 I. 
 
 Of Geometrical Definitions and Rudiments. 
 
 Plate I. 
 
 Point in the praftice of geometry, is the 
 leaft fupcrficial appearance as can be 
 made by the point of a pen, pencil, pin, 
 eSfc. as the point A, and is to be divided 
 by the mind, tho’ not by the hand, into 
 any number of parts, as is conceived, FJg i- 
 notwithftanding that Euclid, and many 
 other fiimous geometricians, has defin’d a point to be nei¬ 
 ther quantity or part of quantity, and therefore not to 
 
 B be 
 

 2 
 
 Fig, 
 
 II. 
 
 Fig.III. 
 
 Fig. IV. 
 
 Fig. V. 
 
 Fig. VII. 
 
 Fig. vn. 
 
 Fig. Vlll. 
 
 Of Geometrical definitions 
 
 be divided into parts; but how 'tis demoiillrated, neither 
 he or any other has fet forth. 
 
 (1.) A line in the praftice of Geometry, is a length, with 
 fuch a breadth, as is given thereunto by the point of the 
 pen, pencil, as defcnbes the fame, which is quite con¬ 
 trary to all other authors, who define a line to be a length 
 without breadth or thicknefs, but without any fort of de- 
 monlfration whatfoever to prove the fame. 
 
 (3.) Of lines there be divers kinds, as' right, circular 
 elhptical, parabolical, hyperbolical, ’ 
 
 (4.) A right line is generated by the point of a pen 
 pencil, ^x. moving from one point to another, the nearelf 
 way; therefore a right line is the nearefl diltance contain’d 
 between two points, as the diflance between the jioints 
 
 A, B. Ihe end or limits of all right lines are points as 
 tlie points A B. 
 
 is generated by the motion of one 
 end of a right line. Suppofe A C to be a right line, fix’d 
 at C as on a center; then by moving it out of the pofition 
 A C to C B, the point A will deferibe or generate the arch 
 or Circular line, A B; and if you move it forward to its for¬ 
 mer pofition A C, the point A will deferibe or generate 
 the circumference of a circle. 
 
 (d.) An elliptical line, or ellipfis, is generated by an ob¬ 
 lique fedtion of a cylinder. 
 
 (7.) A parabolical line is generated by a parallel fedion 
 of a cone. As alfb a hyperbolical curve, the former to 
 the fide, and the latter to the axis. 
 
 (8.) As points terminate lines, fo do lines fuperficial 
 hgures. 
 
 (9.) A fuperficial figure hath length and breadth only, 
 and is contain’d under one termination or many So 
 A is contain’d under one line or termination, B under 
 tw^o, C under three, D under four, E under five, ^c. 
 
 (10.) A circle is a plain geometrical figure, contain’d un¬ 
 der one line, called the periferie or circumference. 
 
 (i I.) Every circumference of a circle is deferibed accord¬ 
 ing to the yth hereof, and the point on which the de- 
 feribent rells, is the center. So in Fig. III. the point C is 
 the center thereof Therefore, as the center of a circle is 
 the exaft midfl of the fame, all right lines drawn flom 
 thence to the circumference, are equal one to the other 
 as in Fig. VII. A B is equal to B C, and that to B D, 
 
 (ii.) The diameter of a circle is a right line drawn 
 thiOLigh the center, and ending at the circumference, as 
 the line ABC. 
 
 3 
 
 (13.) The 
 
3 
 
 and Rudiments. 
 
 (15.) The radius, or femidiameter of a circle, is half 
 the diameter. 
 
 (14.) A fedlion, fegment, portion, or part of a circle, is 
 a figure contain’d under one right line, and part of the tig- ix. 
 circumference. So the right line A B divideth the circle 
 into two unequal parts, and are the fedtions, fegments, 
 portions, or part of that circle. 
 
 (15-.) A femicircle is one half of a whole circle, as the -n 
 
 figure A. / 
 
 (id.) A quadrant is one half of a femicircle, as the fi-v Fig. x. 
 
 gure B. . . \ 
 
 (17.) The radius of a quadrant, is either of the ftreightJ 
 fides, as n m, or m 0, and the circular fide n 0 is called 
 the limb, which is always divided into degrees and min. 
 as will hereafter be fully fliewn in its proper place. 
 
 (18.) An ellipfis is alfo a plain geometrical figure, con¬ 
 tain’d under one line, called the circumference, and is 
 generated according to the 6th hereof; and as the diame¬ 
 ters of a circle are equal to each other, fo likewife 
 arc the diameters of one ellipfis to another, when both 
 are of the fame dimenfion, but at no other time. There¬ 
 fore in ellipfis’s there is a great variety contain’d. 
 
 (19.) Every ellipfis hath two diameters, the one longer 
 than the other ; the longeft diameter is called the conju- 
 gate diameter, and the fliortell the tranfverfe diameter ; 
 the point of interfedfion of both diameters as A, is the 
 center of the ellipfis. 
 
 (zo.) A triangle is a geometrical figure contain’d under 
 three fides, and is either right lined as the triangle A, or rig. xir. 
 circular as C, or mix'd as B. 
 
 (z I.) When a geometrical figure confifts of four fides and 
 angles, and all equal as the figure B, fuch a figure is 
 called a quadrat, or geometrical fquare ; but if of the 
 four fides, two be longer than the other, each to its cor- 
 refpondent, and the angles equal as the figure C, ’tis called 
 an oblong, long-fquare, or parallelogram ; alfo when the 
 fides be all equal, and the angles unequal, as the figure D, Fig. xiii. 
 fuch a figure is called a rhombus or diamond form ; but if 
 fuch a figure fliould have two fides longer and two fliorter, 
 each to his oppolite correfponding, as the figure E,’tis called 
 a rhomboyades ; and when the fides are all unequal, and 
 the angles the fame as the figure F, fuch a figure is called 
 a trapezium. 
 
 (zz.) When any figure contains more than four unequal 
 fides, and angles, fuch are in general called irregular fi¬ 
 gures. 
 
 (zg.) When 
 
4 
 
 I 
 
 Fig. XIV. 
 
 Fig. XV. 
 
 Fig. XV. 
 
 Fig. XVI. 
 
 Of Geometrical Definitions 
 
 (a5,) When a geometrical figure contains five equal 
 fides and angles, as the figure A, llich a figure is called a 
 pentagon ; and if fix as 11, a Jicxagon ; if leven as C, a 
 heptagon ; if eight as D, an odtagon ; if nine as E, a nona- 
 gon ; and if ten as F, a decagon. 
 
 (7-4.) The diagonal lines of a geometrical fquare, are two 
 right lines, drawn Ifom one angle to the other, as tlie lines 
 A B and CD. 
 
 The diameters of a geometrical fquare, are two 
 right lines drawn through the interfedfion of tlic diago¬ 
 nals, parallel to tire fidcs of the fquare, as the lines EF 
 and I K. 
 
 {z 6 .) The center of a geometrical fquare, is a point ,pf 
 intcrfeclion of tlic diagonals, or diameters, or both, it 
 being the iamc as the point L. And what is here 
 liud of a geometrical fquare, the liimc is to be undcrllood 
 ot an oblong, or parallelogram, rhombus, rhomboyades and 
 trapezium. 
 
 (xy.) As lines terminate fuperficial figures, fo do fu- 
 pcrficial figures folid bodies. 
 
 (x8.) A folid body hath three dimenfions, viz. length, 
 breadth, and (thicknefs or) depth. 
 
 (xci.) Geometrical folid bodies, arc the fphcrc, 
 fpheriod, cone, frulhim of a cone, cylinder, pyramis, Iruf- 
 tum of a pyramis, prifm, tetraedron, frultum of a tc- 
 traedron, cube, frullum of a cube, parallelcpipcdon, 
 odfaedron, dodecaedron, and icofaedron. 
 
 (50.) A Iphere, globe, or ball, is generated by the re¬ 
 volution of a femicircle, about its own diameter. So 
 alio is a fpheriod, by the revolution of a femi-ellipfis on its 
 longeft diameter, as figure A and B. 
 
 (3 I.) A cone, is generated by tJie revolution of a right 
 angled tilain triangle about one of its legs, as the figure 
 D. So alfo is a cylinder by the revolution of a paralle¬ 
 logram about one of its lides, as the figure Fh 
 
 (3x.) The fruftum of a cone, is the remains of a cone, 
 when a part thereof is taken away from the upper part, 
 as F L G, taken away from H L I, leaves the frultum 
 F G FI I ; and what is here faid of the frultum of a cone, 
 the fame is to be underftood in the frultum of a py¬ 
 ramis or pyrament. 
 
 (33.) A pyramis, or pyrament, is a folid, which hath 
 a triangle, fquare, polygon, &c. for its bafe, and hath as 
 many reclining faces as are fides contain’d m the bafe, which 
 all terminate m a point like a cone, which point, or ter¬ 
 mination, is called the vertex, or vertical point of the 
 
 pyrament 
 
and Rudiments. 
 
 pyramcnt or cone. See figure K, which is a pyrament, 
 whofe bafe is a geometrical fquare. 
 
 (tx.) a prifm is a Iblid body of five faces^ three of 
 W'hich are parallelograms, and two equilateral triangles, 
 as the figure M. 
 
 (33.) A tetraedron is a folid, containing four faces, 
 each an equilateral triangle, and is one of thofe five bo¬ 
 dies, as are called, the regular, or platonick bodies, as 
 the figure N. 
 
 (34.) The fruftum of a tetraedron is a tetraedron with 
 the angles or vertexes cut off, or a fmall tetraedron cut 
 from every angle. Thisbody thus cut, is compofed of eight 
 faces, wis. four hexagons, and four equilateral triangles, and 
 is as agreeable a body as any herein contained. See fi¬ 
 gure O. 
 
 (3J-.) A eube is a folid body containing fix faces, each 
 a geometrical fquare, as figure P. 
 
 (3d.) The fruitum of a cube, is a cube with the angles cut 
 off, or’tis a cube, that has had a pyramis cut from each an¬ 
 gle, this folid contains fourteen faces, of which fix are oc¬ 
 tagons, and eight equilateral triangles, which being taken 
 together is a very handfome body. See figure There 
 is alfo another body, as is not a great deal different from 
 the preceding, which by workmen is called the canted 
 cube, and is no other than the greateft pyrament, as can 
 be taken from each angle, (v'hich in the former was 
 not.) This body thus cut, contains the fame number of 
 faces as the preceding; but inltead of having fix oftagons 
 and eight fmall triangles, it hath fix geometrical fquares, 
 and eight very large equilateral triangles. See fi- ri 
 gure U. 
 
 (37.) A parallepipedon is a folid body, containing fix 
 fiices (as the cube) whereof but two are geometrical 
 fquares, and the other four, parallelograms ; but a paral¬ 
 lepipedon may have all its faces parallelograms, when its 
 ends are parallelograms, inftead of geometrical fquares. 
 See the figures R and S. 
 
 (38.) An odlaedron is a folid body, eontaining eight 
 faces, each an equilateral triangle. 
 
 (39.) A dodecaedron is a folid body, containing twelve 
 faces, and each a pentagon. 
 
 (40.) An Icofaedron is a folid body, containing twen¬ 
 ty faces, and each an equilateral triangle. 
 
 (41.) The balls of a fphere, or fpheriod, is but a 
 point. 
 
 (41.) The balls of a evlinder is a right line. 
 
 ' C (43.) The 
 
 5 
 
 g. xvr. 
 
6 
 
 Of 
 
 Geometrical Vrohlems. 
 
 (43.) The bafis of a cone is a circle. 
 
 (44.) Befides the preceeding folids there be two others, 
 viz. one of twelve faces, and another of thirty, and every 
 one a rhombus or diamond form. And as thefe definitions 
 are full fufficient for any furveyor, I fliall now proceed to 
 the fecond lecdion. 
 
 Sect. II. 
 
 Of Geometrical Problems, 
 
 Problem I. 
 
 Fig. XVII. 
 
 'T'O divide the right line A B, into two equal parts, ly 
 tlje perpendicular d d. 
 
 Open your compalles to any diftance, that is more than 
 half the line A B. Place one foot or point in A, and with 
 the other defcribe an arch as e e, then with the lame 
 opening on B, defcribe the arch c c, which will interfedl; 
 the firll arch e e,\\\ d d\ draw a right line from d to d,t\\.c 
 two interlctlions, and it lhall divide the given line A B, 
 into two equal parts, in the point E, and fliall be perpen¬ 
 dicular thereunto. 
 
 A perpendicular is a right line, erected upon a right 
 line, making the angles equal on each fide, as E d, on 
 either fide A B. 
 
 Ufe. 
 
 This problem is of great ufe in the fetting out of build¬ 
 ings and gardens, as well as in drawing or defigning the 
 lame on paper. In the praftice of which, a ten foot rod, 
 or a garden line, fupplies the place of compalles, for to 
 defcribe the arches of interfedlion. 
 
 Problem II. 
 
 Upon any point as E, given in the right line A B, to e- 
 rect the perpendicular I E. 
 
 I. Open your compalles to any fmall diftance, and 
 placing one foot in the given point E, with the other 
 !()ot interfecl the given line on each fide, as at c and d. 
 
 1. Open your compafles to any greater diftance, and 
 placing one point in d, with the other defcribe the arch 
 a hh-. 
 
Of Geometrical Trohlema. 7 
 
 I b ; alfo with the fame opening on the point e, defcribe 
 
 the arch a a, interfering the firft arch b b, in the tig-xviii, 
 
 point L 
 
 3. Draw the line I E, and it lliall be the perpendicu¬ 
 lar required. 
 
 Ufe. 
 
 This is alfo a very ufeful problem, as alfo are all the 
 enfuing, both in building and gardening, in dividing ol 
 the parts thereof, which are too numerous to be infert^ 
 ed here, and therefore are omitted till a more convex 
 nient time, when I llrall prefent the world with a parti¬ 
 cular difcourfe on that fubjefl; for the inftrurion of 
 Inch youth, whofe natural genius tends either to archi- 
 tefture or gardening. 
 
 Problem TIL 
 
 From ihe end of the right line AC, C, to ere 61 the 
 perpendicular C D. 
 
 I. Open your compalles to any diftance, and let one 
 foot in C, defcribe the arch B, n, m, and upon it fet 
 the fame opening from B to and from n to m. Fig, xix. 
 
 a. With the fame diftance, or opening of your compaf- 
 fes, defcribe the arch n f, on the point m, and alfo the 
 arch e m, on tlte point n, interfering the arch n f, in the 
 point D. 
 
 3. Draw the right line C D, and it lliall be the perpen¬ 
 dicular required. This problem may be performed many 
 other ways ; but none better or eafier than the pre¬ 
 ceding and the following. 
 
 Problem IV. 
 
 How to ere 6 t a perpendicular upon the end of a line, af¬ 
 ter another manner. 
 
 I. With any opening of the compalfes, defcribe the 
 arch B g, on the point C, and fet that opening from 
 B to g. 
 
 X. Defcribe the arch B D E F, on the point g, with the 
 fame opening as before ; and upon this arch fet up the 
 fime opening three times, viz. Irom B to D, from D to E, pjg. xx. 
 and from E to F. 
 
 3. Draw a right line from F to C, and it lhall be the 
 perpendicular required. 
 
 Problem 
 
Of Geometrical Problems. 
 
 Problem V. 
 
 To let fall a perpendicular line, frotn a point to a rivht 
 line given. * 
 
 ^ the performance of this problem, there is two 
 
 point is over or near 
 tlie middle of the line. And the fecond, when near or 
 over the end of the line. 
 
 Cafe I. 
 
 P Si^^en, and from the point 
 
 1 to let fall the perpendicular P Q, ^ 
 
 I. Open your compall’es to any diltance greater than 
 
 r-. XXI ’ no^;,r R S, interfecA- 
 
 tig. XXI.^ mg the given line in the points R and S. 
 
 z. With any opening on the point R deferibe the arch 
 
 V V, and with the fame opening on the point S deferibe the 
 arch m m, interfering the firft arch, in the point I. 
 
 V n ^ ^ line 
 
 I 0., and It will be the perpendicular required. 
 
 Cafe II. 
 
 w ‘'•"d from the point 
 
 V to let fall the perpendicular V M. 
 
 Fi.xxii r From the given point V, to any part of the given 
 »■ line T O, draw a right line as V N, and by the firft here¬ 
 of divide it into two equal parts in the point X. 
 
 On the point X with the diftance V X or X N, de- 
 fcribe the arch or femicircle V M N, interfering the m- 
 ven line in the point M. 
 
 3. From the point given, to M the interfered point ■ 
 draw the right line V M, and it fliall be the perpendicu¬ 
 lar required. 
 
 Problem VI. 
 
 To deferjhe a right line, parallel to a right line at any 
 dijtance ajjigned. ^ 
 
 Definition. 
 
 Parallel right lines are fuch, that being infinitely con¬ 
 tinued would never meet. 
 
 01 
 
9 
 
 Of Geometrical 'Prohlems 
 
 of parallel lines there be principally two kinds, viz. 
 right lined parallels and circular parallels, as in the fol¬ 
 lowing problems. 
 
 In defcnbing of right lined parallels, there are two cafes; 
 the firft, to draw a right line parallel to a right line at 
 any diftance given ; the other, thro’ a point affign’d, which 
 point n:ay be over, under, or oblique to the given 
 line. 
 
 Cafe I. 
 
 LetEF be a right line given, and let it be required 
 to draw another right line parallel thereunto, at the dif- rig- xxm- 
 tance of G H. 
 
 I. Take in your compaffes the given line G H, and on 
 any part of the given line E F, as at E, defcribe the arch 
 i k, as alfo towards the other end, as at F, with the 
 fame diftance, defcribe the arch c m. 
 
 tl. a line drawn by the convexity of thofe two arches, 
 fliall be the parallel required, at the parallel diftance 
 of G H. 
 
 Cafe II. 
 
 Let A B be a right line given, and let it be required to 
 draw another right line parallel thereunto ; that fliall pafs 
 thro’ the point E. 
 
 I. Take with your compafles the neareft diftance from 
 the given point E, to the given line A B, and with that rig. axiv. 
 diftance, at the end A, defcribe the arch n n. 
 
 A right line drawn through the given point E,by the 
 convexity of the arch n n, fliall be the parallel delired, at 
 the parallel diftance of the given point E. 
 
 Problem VII. 
 
 To make the angle M C B, eqtial to the given angle 
 E AN. 
 
 1. Upon the angular point A, with any opening of the 
 compafles, defcribe the arch o o, and with the fame open- pig. xxv. 
 ing fet one point, or foot of the compaffes, on the point 
 
 C, and defcribe the arch n n. 
 
 2. Take the diftance o o, and fet it from n to n. 
 
 3. A line being drawn from C to n, fliall make the 
 angle M C B, equal to the angle E A N, as required. 
 
 This problem is of great ufe in taking the plan of build¬ 
 ings, gardens, &c. 
 
 D 
 
 Problem 
 
Of Geometrical Problems. 
 
 Problem VIII. 
 
 To dhide an angle, A B C, into two equal 
 ■parts. 
 
 I. Upon the angular point B, with any opening, de- 
 fcribe the arch r r, interfering the fides of the angle in 
 the points r r. 
 
 Fig. XXVI. -i. With any opening, on the points r r, defcribe the 
 arches m m and v 'v, interfering each other in the 
 point L. 
 
 5. A right line drawn front L to B, fltall divide the an¬ 
 gle A B C, into two equal parts as required. 
 
 Problem IX. 
 
 T0 divide a right line into any number of equal parts. 
 
 I.et it be required to divide AI N into fix equal parts. 
 
 I. From the end M or N, draw a right line at pleafure, 
 as A M. 
 
 a. Make the angle N ME equal to MNA, by Prob. VII. 
 
 Fig.xx\ II. fecond cafe of Prob. VI. make M E parallel 
 
 to A N. 
 
 3. Open your compafl'es to any fmall dillance at plea¬ 
 fure, and fet off that dillance five times from N towards 
 A, and from M towards E, as at the points i, %, 
 DJ 4^ S - 
 
 4. Draw right lines from y to i, from 4 to x, from 
 5 to 3, from X to 4, and from i to y ; and their interfebli- 
 ons will divide the given line M N, into fix equal parts, 
 as required. 
 
 Problem X. 
 
 T0 find a mean proportion between two right lines gi¬ 
 ven. Let it be required to find a mean proportion, be¬ 
 tween the given lines N and O. 
 
 I. Make A D, equal in length to both the lines O and 
 N, and by Problem I. divide it into two equal parts, in 
 the point C. 
 
 X. On the point C, defcribe the femicircle, making the 
 F. XXVIII. diameter equal to A D. 
 
Of Geometrical Problems. ir 
 
 g;. At E (the joining of both lines) ere£t the perpen¬ 
 dicular E h and continue it till it meet the curve in the 
 point I. 
 
 4. The line E I is the mean proportion required. 
 
 Problem XL 
 
 To find the center of a circle as Jhall pafs through anj 
 three points given, as are not in a right line. 
 
 Let the three given points be D B A. 
 
 I. Draw a right line from any one of the points, as 
 A, to either of the other points, as to B, and alfo draw 
 another right line from B to D. 
 
 a. Byprobleml. divide thofe two equal parts by two per- Fig.xxix. 
 pendiculars, as the perpendicular lines H F and C E, 
 which perpendicular lines do always interfeft each other, 
 and the point of interfedlion is the center of a circle as 
 Will pals through the points alligned. 
 
 Problem XII. 
 
 To inferibe a triangle geometrical fquare, pentagon, 
 hexagon, heptagon, oCtagon, nonagon, or decagon, 'luithin a 
 circle, 
 
 I. Delcribe the circle A F C G, and draw the diameter 
 A C and F G, interfering each other at right angles, in 
 the center E. 
 
 а. Make A B and AD, equal to the femidiameter EC, 
 and draw the right line B D, wliich is the fide of an equi¬ 
 lateral triangle, as may be inferibed in that circle. 
 
 3. Draw the right line A F, and it lliall be the fide of a 
 geometrical fquare. 
 
 4. Upon H, with the diftance H F, deferibe the arch 
 F I, and draw tlie right line F I, which is the fide of a 
 pentagon as may be inferibed therein. The diameter 
 
 A C, or FG, is the fide of a hexagon, and half BD; as Fig.xxx. 
 H B, or H D, is the fide of a heptagon or feptagon. 
 
 S- From E, the center through M, draw the right line 
 E M K, fo fhall the diftance, or right line A K, be the 
 fide of an odlagon. 
 
 б . Divide the arch BAD, into three equal parts, each 
 of which is the fide of a nonagon, as D S. 
 
 7. The diftance E I is the fide of a decagon. Every 
 fide in the figure is number d witli its proper number, 
 
 as 
 
i z Of Geometrical Problems 
 
 as the fide of a pentagon with number y, a hexagon with 
 the number 6 , &c. 
 
 This figure, thus made, is a very ufeful inftrumcnt to 
 infcribe any poligon in a circle, when required ; as for 
 example : 
 
 Le/ it he required to infcribe a nonagon in the circle 
 A Ji C D. 
 
 Fig.XXXI. I. On the center E, defcribe the circle F, G, H, I, e- 
 
 qual in diameter to the circle A F C G, fig. XXX. 
 
 From thence take the diftance S D, and fct that dif- 
 tance from F to V, from V to Q^, from to P, &c. to 
 the point F, where you began. 
 
 3. Lay a ruler from the center E, to the feveral points 
 F V CFP, &c. and ’twill cut the outer circle in the points 
 X X X, &c. 
 
 4. Draw lines from x to x, &c. and thofe lines lhall 
 form the nonagon required. And what is here faid of a 
 nonagon, the fame rule is to be underltood of any other 
 figure, as before defcribed. 
 
 Problem XIIl. 
 
 To make an equilateral triangle, as A, B, O, whofe 
 fides Jhall he equal to any given line, as the right line 
 N M. 
 
 I. Make A B equal to the given line N M, and with 
 the diftance A B, on the point A, defcribe the arch v v, 
 and with the fame diftance on the point B, defcribe the 
 arch a a, interfefting the arch v v, in the point O. 
 
 F XXXII interfedlion O, the right lines A O, 
 
 and B O, and they will complete the equilateral triangle 
 whofe fades are each equal to the given line N M, as re¬ 
 quired. 
 
 Problem XIV. 
 
 Three unequal right lines, R S T, being given to 
 make a right lined triangle, whofe fides Jhall be equal 
 thereunto. 
 
 I. Make A B, equal to R. 
 
 X. Take the line S in your compaflcs, and on A de¬ 
 fcribe the arch a a. 
 
 3. Take 
 
 I 
 
Of Geometrical VroUems. 
 
 ^3 
 
 5. Take the line T in your compalles, and on B de- 
 fcribe the arch m m, interfering the firll arch in the 
 point C. 
 
 4. Draw from the interfeflion C, the right lines C A 
 and C B, and they will complete the triangle, whofe lides 
 are refpeftively equal to the given line-^ R S T, as re¬ 
 quired. 
 
 A B’) CR 
 
 A Cy equal to the given line<S 
 
 T 
 
 Problem XV. 
 
 To deferibe a geometrical fquare, whofe fides JImU be 
 equal to a right line given. 
 
 Let it be required to make the geometrical fquare 
 M N O P, whofe lides fliall be relpecTively equal to the 
 given line A B. 
 
 I. Make O P equal to A B. 
 
 1. On the point P ere< 3 t the perpendicular P N, (by 
 problem III, or IV, hereof ) and make it equal in length 
 to the given line A B . 
 
 3. With the diftance A B, on the point N, deferibe the 
 arch n n, and with the fame diftance, on the point O, de¬ 
 feribe the arch a a, interfering the former in the 
 point M. 
 
 4. From M the point of interferion, draw the right 
 lines M N and M O, and they will complete the geome¬ 
 trical fquare, as required. 
 
 Problem XVI. 
 
 To mahe an oblong parallelogram, or long fquare, as 
 A B C D, whofe length and breadth JJjall be equal to two 
 given lines, as N O. 
 
 Fig. 
 
 I. Make the line C D, equal to the given line O, and 
 on D, (by the Illd problem hereof) erer the perpendicular 
 B D, and make it equal to the given line N. 
 
 X. On B, with the diftance C D, deferibe the arch 0 0, 
 and on the point C, with the diftance B D, deferibe the 
 arch r r, interfering the former in the point A. 
 
 n E 3. From 
 
 
5- From A the point of interfeftion, draw the right 
 lines A B and A Q and they will complete the oblons 
 as required 
 
 Fig. 
 
 XXXVI. 
 
 ine nue n rr ana u U) ■ w ..i • i ,■ r(J 
 
 The fide A C and B 
 
 Problem XVII. 
 
 To 7 nake a rhomhus, or diamond form, whoje fides 
 fjall le equal to a right line given. 
 
 ^ Let it be required to defcribe the rhombus A, M, 
 N, Oj whofe fides Ihall be each equal to the given 
 line V, R. 
 
 I. Make A O equal to V R, and on the point O, with 
 the dillance OA, defcribe the arch A MN. 
 
 1. With the fame dillance fet up the opening of the 
 compalles from A to M, and from M to N. 
 
 3. From the point A to the point M, draw the right 
 line A M, and from the point M, draw the right line 
 M N to the point N ; and laftly, draw the line NO, and 
 you will complete the rhombus as required, with its re- 
 ipedtive fides equal to the given line V R. 
 
 jir 
 
 Fig. 
 
 XXXVll 
 
 Problem XVIII. 
 
 To make a rhomhoyades, whofe fides fall he equal to 
 tnvo given right lines, as L and Q_; and the acute 
 angles at M and O, equal to the given angle Z. 
 
 I. Make P M equal to the given line L, and by pro¬ 
 blem \TI, make the angle E M P, equal to the angle Z, 
 anci make M E equal to the given line Q_. 
 
 1. Take in your compalles the given line L, and on E, 
 defcribe the arch n n. 
 
 3. Take the length of the other given line CL, and on 
 the point P, defcribe the arch a a, interfedling the for¬ 
 mer in the point O. 
 
 4-. From O the point of interfcdion, draw the lines 
 O E and O P, and they will conllitute the rhomboyades 
 as required, whofe fides O E and P M, Ihall be equal to 
 the given hue L, and the lides O P and E M, equal to 
 the given line (L, as alfq the angles at O and M, equal 
 to the given angle Z. 
 
 Problem 
 
Of Geometrical VroUemr. u 
 
 Problem XIX. 
 
 To make a trapezium {as the figure R N O M) luhofe 
 fides Jhall he equal to four right lines given, as the lines 
 D, E, V, T;, and one angle, as the angle N, equal to an 
 angle given, as the angle Z. 
 
 I. Make N M equal to the given line D^, and by pro¬ 
 blem VII. make the angle at N, equal to the given angle 
 T, and make N, R, equal to the given line E. 
 
 r. Take the given line V in your compalles, and on M 
 defcribe the arch n n, then take the given line T, and on xxxviii 
 R delcribe the arch a a, interlebling the former arch in 
 the point O. 
 
 3. From the point of interfedion O, draw the right 
 lines O M and O and they Ihall complete the trapezi¬ 
 um, as required, with its refpeblive fides equal to the 
 lines given. 
 
 Problem XX. 
 
 How to dejcrihe an elUpfis to any lerigth and breadth gi¬ 
 ven, as the [figure A B C M, nHjofe longefi diameter is e- 
 qualto the given line DV, and the JhorteJlto the Ime EP. 
 
 I. Make the right line A C equal to the given line D V, 
 and (by problem I.) divide it into two equal parts, by 
 the perpendicular B M, which make equal to the uiven 
 line E P. u 
 
 a. Take half the longefi: diameter, as A F orC F, and on 
 B defcribe the arches a a, and a a, interfedling the longefi 
 diameter in the points O and N, whicli are the two cen¬ 
 ters by which the ellipfis may be defcribed. 
 
 3. Fallen two pins, or tacks, (if on the ground, as in a 
 garden, two fiakes) at O and N, and putting a line about 
 them, fallen the ends together, at the length of tlie line 
 O C, or N A, fo that the firing may move about both the 
 pins, tacks, efrc. at pleafure. 
 
 4. Take a black-lead pencil, tracer, ^c. and extending 
 the line therewith, it will, by its motion about thole 
 two centeis, delciibe an ellipfis, as Ihall be equal in length 
 and breadth to the given lines D V and E P,’ as re¬ 
 quired. 
 
 XXXIX. 
 
 4 
 
 Problem 
 
■WM 
 
 to dcJcTtho UH clhpjis to (itiy length ciud hveudth^ 
 m the figure A B C 'luhoje longeji diameter is equal 
 
 to the given line and the fiorte/i to the line hy the 
 help of a pair of compaffes, without the ajjifiance of a 
 line and tracer, as in the preceding proUem. 
 
 Fis;. XL 
 
 I. Defcribc the longeft and fliorteft diameters^ equal to 
 the given lines, interfering each other, at right angles 
 in the point E (as in the preceding). 
 
 1. Take half the fhortelt diameter, as B E, and place 
 that diftance from A to F on the longeft diameter. 
 
 _ Divide the fpace between F and E the center, into 
 three equal parts, and place one of thofe parts backward 
 from F to I. 
 
 ^ 4. Make E K equal to E I, and on K, with the diftance 
 K 1, defcribc the arches n n on the one fide, and n n on 
 tire other. 
 
 T- With the fiime diftance, on the point I, defcribc the 
 arches 0, 0, and 0, 0, interfering the other two in the 
 points L and M. 
 
 6 . Lay a ruler from L to I, and draw the line IV; alfo 
 from L to K, and draw the K P ; alfo from M to k’, and 
 diaw the line K and alfo from M to I, and draw the 
 line I R. 
 
 7. OiM, with the diftance, IA deferibe the arch z A z, 
 and on K, with the fame opening, deferibe the arch x Cx. 
 
 8. On M, with the diftance M 2:, deferibe the arch 
 
 B A’ ; and on L, with the fame opening, deferibe the 
 
 ai ch X D z, and thus is the elliplis completed, as 
 required. 
 
 Problem XXII. 
 
 To inferihe a circle niithin a fquare. 
 
 I Draw the diagonals N S and V M, interfering each 
 other in the point O. 
 
 a. From the point O, let fall the perpendicular O C, 
 and with the opening O C on O, deferibe the circle as 
 required. 
 
 Problem 
 
Of Geometrical VroUems. 
 
 
 Problem XXIII. 
 
 To infcribe a fquare within a circle, and to circumfcrihe 
 a circle about a Jquare. 
 
 I. By the third of problem XII. infcribe the fquare 
 A E I Oj and draw the diagonals A O and E I, inter- tig- xlii. 
 fefting each other in the point N. 
 
 X. On N, with the diftance N A, N E, N I, or N O, 
 
 (they being all equal to each other, by definition ii. fig. 
 
 Vn.) defcribe the circumfcribing circle, as required. 
 
 Problem XXIV. 
 
 To infcribe a circle within a triangle, as the circle 
 M, o, e, within the equilateral triangle A E N. 
 
 I. Divide any two of the angles of the given triangle, 
 as the angle N and A, into two equal parts, by the lines . 
 
 N 0 and A e, interfecling each other in the point M, (as " 
 by problem VIII. hereof). 
 
 X. From M let fall the perpendicular M P ; and on M, 
 with the diftance M P, delcribe the infcribed circle, as 
 required. 
 
 Problem XXV. 
 
 To circumfcribe a circle about a triangle. 
 
 The folution of this problem is exaftly the fame as 
 problem XL For if you fuppofe the three angular points . , 
 
 B C A, to be three given points, as in that problem, 
 the operation hereof is exadlly the fame, and therefore 
 needs no further demonftration. 
 
 Problem XXVI. 
 
 To find the center of a circle, that fall pafs through 
 any two given points nvithin a circle, and divide the cir¬ 
 cumference into two equal parts. 
 
 Let M N be the given points. 
 
 F 
 
 I. From 
 
1 8 Of Geometrical Problems. 
 
 1. From any one of the points as M, draw a riglit line 
 tirrough the center O, extending it infinitely to yE. Up¬ 
 on this line, at the center O, ereU the perpendiculav 
 tig. XLv. O W, and from W through M, draw the line W R ; and 
 from R, through O the center, draw the diaineter 
 R, O, a. 
 
 z. Draw the right line W a, and extend it till it inter- 
 fecT: the line M O JE., in the point V ; through which, 
 and the given points M and N, you may delcribe the 
 arch of a circle (by problem XL) as will divide the cir¬ 
 cumference given into two equal parts, and pafs through 
 the two given points, as required. 
 
 Problem XXVII. 
 
 T0 make a geometrical fquare, as A E M N, equal 
 in area to any right lined triangle, as the triangle I O M, 
 given. 
 
 I. Let fall the perpendicular O R, and make M S e- 
 qual to half the perpendicular O R. 
 lig. XLVi. Divide I S into two equal parts at R, and on R, 
 with the diftancc I R, or R S, defcribe the lemicircle 
 IAS. 
 
 5. At M creel: the perpendicular M A, and extend it 
 till it interfedb tltc femicircle in A. 
 
 4. The line A M is the fide of a geometrical fquare, 
 whofe area is equal to the area of the triangle given, as 
 required. 
 
 Problem XXVlll. 
 
 T0 make a geometrical fquare, equal to a parallelogram 
 given. 
 
 Let it be required to jnake a geometrical fquare equal 
 in area to the oblong, or parallelogram, A B C D. 
 Fig.XLvii. Continue the fide C D to F, making D F equal to 
 BD, and divide C F into two equal parts at G, and there¬ 
 on, with the diftance G C, or G F, defcribe the arch 
 C E F. 
 
 x. Continue DB to E,and then will D E be a mean pro¬ 
 portional, and the fide of a geometrical fquare, whofe 
 area is equal to the oblong, or parallelogram, A B C D 
 given, as required. 
 
 Problem 
 
Of Geometrical Trohlems. tp 
 
 Problem XXIX. 
 
 To divide a line given, in Juch proportion as another 
 is before divided. 
 
 Let it be required to divide the right line M, in fuch 
 proportion as the line A E. 
 
 I. By problem XIII. hereof, make the equilateral tri¬ 
 angle G H I, with its fides equal to the line A E, and di¬ 
 vide any one lide thereof, as H I, in the fame proportion, 
 as A E (the length being equal). 
 
 a. Take the length of the line M, and fet it from G (the rig. 
 angle oppofite to the lide divided) to V, on one fide, and xlviii. 
 to O, on the other fide, and draw the right line V O. 
 
 3. Lines being drawn from G,thro’ the points i, a, 3, 
 
 4, y, and 6 , lliall interfedl the line V O, in the points 
 000, &c. and divide that line in the very fame pro¬ 
 portion as the given line H I, as was required. 
 
 Problem XXX. 
 
 To divide the circumference of any circle into 36^0 epial 
 parts, as the circle A B C D, fig. XLIX. 
 
 I. Draw the diameter AC, and by problem I. divide 
 it into two equal parts by the perpendicular B D, then 
 will the circle be divided into four equal parts, and confe- 
 quently tlie circumference alfo. 
 
 1. Open your compalfes to half the diameter, as P A, 
 
 &c. and fet that diftance, firjt, from A to e, and from 
 A to /; fecondly, from B to m, and from B to /; third- 
 ly, from C to h, and from C to 2 ; lafily, from D to h, 
 and from D to g 5 and thus you have divided the ciicum- 
 lerence into ix equal parts, each reprefenting 30 de¬ 
 grees. 
 
 3. Divide each of thofe divifions into three equal parts, 
 and each of thofe parts into ten, and then will the ciicle 
 be divided into 360 equal parts, which are called de¬ 
 grees. It is to be obferved herein, tliat the femi-diame- 
 ter, which is generally called the radius, is always equal 
 in length to 60 degrees, or equal parts of the circumfe¬ 
 rence. Every equal part (or degree) of the circumfe- Fig- xlix- 
 rence, is always fuppofed to be divided into do Iclier e- 
 
 I ' qua! 
 
20 
 
 Of Geometrical Problems. 
 
 qual parts, and thofe are term’d or called minutes. There¬ 
 fore when we mention two degrees and a half^ we fliy two 
 degrees 50 minutes ; or one deg. and we fay one degree 
 and zo min. and when we write down any number of de¬ 
 grees and min. as thirty degrees fifty feven minutes, we 
 write them thus 50° : X7', &c. And what is here faid 
 in the divifion of the circumference of this circle, tlie 
 fiune is to be underllood in the divifion of the circumfe¬ 
 rence of every circle, for in the circumference of every 
 circle, there is always the fame number of degrees there¬ 
 in, although fome circles may be fmaller, and others lar¬ 
 ger than the given circle A E C D. 
 
 Dcmonftration. 
 
 I. Draw right lines from the center P, through every 
 tenth degree of the circumference, and e.'-tend them in¬ 
 finitely. 
 
 z. On P the center, deferibe the inward circle, and the 
 lines before drawn tlirough every tenth degree, will inter- 
 fefl that circle in the points n n ii, &c. and will divide 
 that circumference into thirty fix equal parts, each repre- 
 fenting 10 degrees. Alfo on P deferibe the outward 
 circle 0000, &c. wherein you may obferve the aforefaid 
 lines of every tenth degree, to divide that circumference 
 Fig-xLix. in the very fame proportion, as the inward circle n n nn, 
 &c. and the given circle A E C D. Therefore let any 
 circle be as fmall as may be conceived, or as large as the 
 greatelt circle as can be fuppofed to bound the univerfe, 
 the number of degrees in each are both equal, and con- 
 fequently the minutes the fiime, though greater or lefler 
 each, in fuch proportion as the circumference of one cir¬ 
 cle hath to another, which is what was to be demon- 
 Itrated. 
 
 K. B. Before the young fludent proceeds any fur¬ 
 ther, let him well underiland this problem, for 
 hereon the whole body of mathematicks depends, 
 as alfo the feveral operations following ; but if 
 he finds any difficulty upon the firll or fecond 
 reading, either of this or any other problem, let 
 him not be difeouraged, ’twill by often contempla¬ 
 ting be made eafy ; for mathematicks, is not to be 
 underllood at once I'eading over, like plays, hiflo- 
 ry, or romances. 
 
 4 
 
 Problem 
 
of Geometrical 'ProhJems] 
 
 Problem XXXI. 
 
 To injerihe an eUipfis 'within an oUong, or parallelo¬ 
 gram, as A B C D. 
 
 I. Draw the two diameters of the parallelograjii, as 
 E G and F H, which fuppofe to be the length and breadth 
 of an elliplis given, to deferibe the fame as if they had p;g 
 not been the diameters of the oblong. 
 
 1. By problem XX, or XXI, deferibe the elliplis 
 E F G H, and it will be the ellipfis inferibed, as re¬ 
 quired. 
 
 Problem XXXIL 
 
 To ereB a perpendicular line hy the help of a ten foot 
 rod {or other meafure equally divided) on the ground, in 
 the Jetting out of a building, garden, &c. 
 
 The proportional numbers contained in a fquare, or 
 right angle, is 3, 4; and s \ or 6, 8, and 10 ; there¬ 
 fore if you would raife the perpendicular D F, from the 
 point D, on the line HD; fet olf fix foot from D to E, 
 and with eight foot of your rod at D, deferibe the arch 
 a a ; and allb with ten foot, deferibe the arch B B, on 
 the point E, and the interfeftion F is the perpendicu¬ 
 lar point required ; or, from E lay a ten foot rod, and 
 from D an eight foot rod, and clofe their ends together, 
 and that lhall be the perpendicular point alfo; and a right 
 line drawn from thence to D, flrall be the perpendicular 
 required. 
 
 This problem may be applied to pradlice on paper, 
 if you ufe a fcale of equal parts, and a pair of compaf- 
 fes inftead of the ten foot rod. 
 

 Sect. III. 
 
 Of Geometrical Axioms and Theorems. 
 
 Axiom 
 
 \F to, or from, equal quantities, ie added or fuhtra£l- 
 ed equal quantities, the fums or remainders will he 
 equal. 
 
 Demonflration. 
 
 I. Draw the two diameters A F and L B, interfeding 
 eacji other in the center N, and then will the angle 
 A N L be equal to the angle B N F, for the arches A B 
 and B F completes a femicircle, as alfo do the arches 
 B A and A L. Therefore the arch B F mull: be equal to 
 the arch A L, becaufe the arch A B continues the lame; 
 and by the fame reafon the angle A N B, is equal to the 
 angle L N F. 
 
 Axiom 
 
 Sluantities equal to a third, are equal to one a- 
 nother. 
 
 Demonflration. 
 
 The alternate angles C and F, are equal to each other, 
 as allb E and D ; for the angle C is equal to the an¬ 
 gle B, and the angle B to the angle F, by the preceding 
 axiom. Wherefore C and F being both equal to B, 
 mull be equal to one another, and the like of E and D, 
 which are both equal to the angles A and G. 
 
 Theorem 
 
I 
 
Of Geometrical Axioms and Theorems.' 
 
 T H E O R E M I. 
 
 If a right line do fall on two parallel right lines, it Fig 
 maketh the oppofite angles equal, and the internal angles 
 on the fame fide, equal to two right angles, or i8o 
 degrees. 
 
 I. The right line P falling upon the two parallel 
 right lines R T and S V, do make the angle D equal 
 to A, and C to B, alfo the angle G equal to E, and H 
 to F. 
 
 X. the angle D with F is equal to two right angles, 
 becaufe F is equal to C (by axiom II.) and C and D to¬ 
 gether are equal to two right angles, or a femicircle ; 
 and lince the angle F is equal to the angle C, therefore 
 F D or EC, are equal to two right angles, or i8o deg. 
 which was to be proved. 
 
 Theorem II. 
 
 If multifarious right lines he inter feBe dip mult ifari-v\%. 
 ous right lines, which are parallel one to the other, the 
 fegments are proportional one to the other. 
 
 Demonftration. 
 
 Let the right lines N O and N M, be interfebled by 
 the fix parallel right lines T T, V V, XX, Y Y, Z Z, 
 and W W; then will the interfegments be proportional 
 one to the other. For if N A be one fifth part of N O 
 N B is likewife one fifth part of N M, and the like 
 of all others. 
 
 Theorem III. 
 
 If four right lines he proportional, that is, as the frftr^. 
 is to the fecond, Jo is the third to the fourth ; the paral¬ 
 lelogram made of the two means (or middle terms) 
 will he equal to the parallelogram made of the ex- 
 treams. 
 
 Demonftration. 
 
 Let the four proportionals be A 14, B id, C ix, and 
 D 8 ; I fay, the parallelogram made of the two mean 
 
 terms. 
 
24 - 
 
 rig. t. 
 
 rig. VI, 
 
 Of Geometrical Axioms and Theorems. 
 
 terms, viz. \6 and is equal to the parallelogram 
 made of the two extreams, viz. and 8. Therefore 
 multiply 16 by ix, and the produ6l is equal to ipx, and 
 alfo X4 by 8, and the produd: is equal to iqx, as be¬ 
 fore. Therefore ’tis apparent, that the parallelogram 
 made of the means, is equal in power to the paralle¬ 
 logram made of the extreams, which was to be de- 
 monftrated. 
 
 Theorem IV. 
 
 If three right lines he proportional, viz. as the firft 
 is to the fecond, fo foall the fecond he to a fourth. 
 The fquare made of the means fall he equal to the oh^ 
 long made of the extreams. 
 
 Demonltration. 
 
 Let the proportional lines or numbers be 4, 8, 8, 
 then will it be as 4 is to 8, fo is 8 to id, and the fquare 
 A E I O of the means, will be equal to the parallelogram, 
 made by the extreams. For multiplying the means 8 by 
 8, the produdl is 64, and multiplying the extreams 4 ^nd 
 id by each other, the produft is d4, and is equal to the 
 produ£l of the means which was to be demonftrated. 
 
 Theorem V. 
 
 In every right angled plain triangle, the fquare made 
 of the hypothenufe, or fide which k oppofite to the right 
 angle, is always equal to the fum of the Jquares made 
 of the legs or fides. 
 
 Demonltration. 
 
 Let N O M be a right angled plain triangle, whofe 
 lides are as follows, viz. the fide N M equal to d, and 
 the fide M O equal to 8, then will the hypothenufe be 
 equal to 10. 
 
 2. If you multiply the fide N M into itfelf, its produdl 
 will be equal to 3 d, the fquare N D M I. 
 
 . 3. Multiply the fide M O into itfelf, and its product 
 will be equal to d4, the fquare M O V C. 
 
 4. Add 
 
25 
 
 Of the Conjiru^ion, See. 
 
 4. Add the area of both fquares together, viz. ^6 and 
 64., and their fum will be equal to too. 
 
 T. Multiply the hypothenufe N O into itfelf, and its 
 produa is 100, which is equal to the fum of the fquares 
 made of the legs before added together, as was to be 
 demonftrated 
 
 Sect. IV. 
 
 Plate. II. 
 
 Of the CanfiruHion of Compound Geometrical 
 
 FIGURES. 
 
 I. General yiXIO MS for tloe proportions of figures. 
 
 Axiom I. 
 
 Hat the length of a pi'oportionable parallelogram be' 
 to the breadth, as three is to two ; therefore if the 
 
 rjT 
 
 length be three foot, the breadth mull be two foot. 
 
 Axiom II. 
 
 When a geometrical fquare hath its lides intercepted 
 with femicircles externally, as A, the diameter of every 
 fuch femicircle mull contain ~ of the fide, on which ’tis 
 deferibed ; and the fame proportion alfo, when at the 
 end of a parallelogram, as B. 
 
 Axiom III. 
 
 When the angles of a geometrical Iquare, or oblong, 
 is cut off by the arch of a circle, the radius of thofe qua¬ 
 drants, or arches, mull be \ the length of the fide of a 
 geometrical fquare, or end of the parallelogram, and the 
 lame proportion is to be obferved when the angles are 
 
 H cut 
 
 Fig. IX. 
 
Confiitution of 
 
 , cut off by a fmall geometrical fquare, as the fig. C cut 
 1 by little fquares, and D by quadrants or arches. 
 
 Axiom IV. 
 
 When the fide of a geometrical fquare, or end of a 
 parallelogram, hath its angles cut off by arches or little 
 fquares, and the ^ remaining be intercepted byafemicircle, 
 as E F ; the arches, or little fquares, mull be firft de¬ 
 ferred, and the diameter of the femicircle muff con¬ 
 tain i of the remaiiier, as the femicircle in axiom U. 
 contains ^ of the whole breadth. 
 
 Axiom V. 
 
 When the angle of a geometrical fquare, or oblong, 
 is cut oft'by a part of a geometrical fquare, and the qua¬ 
 drant of a circle, as fig. G ; the radius of thofe arches, 
 or quadrants, mult contain \ of the fide of each little 
 fquare. 
 
 Axiom VI. 
 
 When a compound figure is circumferibed by a com¬ 
 pound figure, thofe arches of the compound figure cir- 
 cumferibing, muff contain j- of that fide to which they 
 belong, fo a, b, contains ^ of CD in figure H ; but the 
 center of all fuch arches muff always be upon the inter¬ 
 nal fig. as at m. 
 
 I O M 
 
 VII. 
 
 When any fide of a right line figure has a fquare breafe. 
 therein, as the figure T at V; the length of that break 
 muff be ^ of A B, {viz. the length of the fide wherein it 
 Hands) and the depth \ ; but when a break happens a- 
 gainft an arch, as at O, in figure T T, thofe breaks 
 muff be made in proportion to the curve of the oppo- 
 fite arch. 
 
 VIII. 
 
 When an arch breaks into an oblong, as the arches 
 m m in figure T T, it muff not break in above of the 
 breadth of the oblong at moft, and the extreams of fuch 
 ' an arch muff ever be five times their depth, They are 
 
 to 
 
27 
 
 Compound Geometrical Figures. 
 
 to be defcribed by problem XI. fedt. I. having the 
 depth, and both extreams given, as three given points. 
 
 Axiom IX. 
 
 When the lides of a geometrical fquare is intercepted 
 with femicircles internally, as figure Z; the diameters 
 of thofe femicircles muft be no more than one halfl, 
 the fide of the fquare, wherein they are defcribed. 
 
 Fig, IX. 
 
 Thefe axioms, and the preceding problems of feft. I. 
 being well underftood, the young ftudent will find no 
 fort of difficulty in defcribing the figures contain’d 
 in the eight enfuing problems, to which we will 
 proceed 
 
 A general Rule concerning Compound Figures. 
 
 ^ H AT every compound (or plain) figure, that is en- 
 compafs d with another figure, be not of the fame 
 kind, yiz. not to incompafs an odlagon with an odlagon 
 but with a circle, or fome other figure as is agreeable 
 thereunto, and the like of all other figures in Ge¬ 
 neral. ° 
 
 Problem I. 
 
 To defcrihe the compound figure A B C D, with its 
 cumjcribing figure E F G H. 
 
 ctr- 
 
 fqiiare® A Tc d” ^ 
 
 a. By axiom IL hereof, defcribe the quadrants of each 
 angle, and thus will the interior figure be completed. 
 
 F F diftance affign’d, draw the fquare 
 
 ^ P^blem VI. fea. I and by axiom VI 
 
 hereof; defcribe the arches o, whofe centers are 
 
 ^uired^* will complete the figure re- 
 
 3 
 
 Problem 
 

 28 
 
 of the Conjlru^ion of 
 
 Problem II. 
 
 To de fcrthe the coynpoimd figure A B C D, with its cir- 
 cumjcrihing figure PI P" G H. 
 
 I. By problem XV. fefl:. 1 . defcribe the fquare ABCD, 
 in Inch proportion as is laid down in axiom 1. hereof, and 
 by axiom III. dediicT: the little fquares from every an¬ 
 gle. 
 
 Fig 11. a. Defcribe the outer parallelogram parallel to the firll 
 at the diliance aflign d, and by axiom VI. defcribe the 
 four arches PI, G, F, E, whofe centers are at a a a a, 
 and they will complete the figure required. 
 
 Problem III. 
 
 To defcribe the compound figure ABCD, with its 
 circumjeribing figure. 
 
 I. By problem XV. fedl. I. defcribe the geometrical 
 Fig' HI- fquare ABCD, and by axiom II. hereof, defcribe the femi- 
 circles, and then by drawing the circumferibing line paral¬ 
 lel thereunto, at any diltance affign’d, the figure is com¬ 
 pleted as required. 
 
 Problem IV. 
 
 To defcribe the compound figure ABCD, with its cir¬ 
 cumferibing figure. 
 
 I. By problem XVI. feft. 1. defcribe the parallelogram 
 Fig. IV. according to the proportion of axiom I. ’ 
 
 a. By axiom III. cut off the angles with the quadrant 
 ol a circle, and by axiom II. defcribe the femicircles at 
 each end, whofe centers are e e. 
 
 Laftly, Defcribe the circumferibing figure parallel there- 
 unto,at any diltance aflign’d ; and the figure is completed 
 as required. ’ 
 
 Problem V. 
 
 To defcribe the compound figure ABCD, with the cir¬ 
 cumferibing figure E F. 
 
 I. By 
 
Compound Geometrical Figures. 
 
 I. By problem XVI. fedl. I. defcribe the parallelogram 
 A B C D, according to the proportion of axiom I. and de¬ 
 fcribe the femicircles at the ends, according to axiom 11. 
 
 a. Defcribe the outward line parallel thereunto at any 
 diftance affigned ; and by axiom VII. defcribe the breaks 
 E F, and the figure is completed as required. 
 
 Problem VI. 
 
 To defcribe the compound figure A B C D, and its cir- 
 cumfcrihing figure E F G H. 
 
 I. By problem XV. fe£l. I. defcribe the geometrical 
 iquare A B C D, and by axiom V. defcribe the angles. 
 
 a. At any affigned diftance defcribe the outer fquare 
 E F G H ; and by axiom VII. defcribe the quadrants at 
 every angle. 
 
 Lafily, By axiom VI. defcribe the arches K M N L, 
 and the figure is completed as required. 
 
 Problem VII. 
 
 To defcribe the compound figure A B C D, and its cir- 
 citmfcribing figure E F G H. 
 
 I. By problem XV. fedl. I. defcribe the geometrical 
 fquare A B C D, and by axiom II. defcribe the femicir¬ 
 cles whofe centers are a a a a and by axiom III. defcribe 
 the arches at every angle. 
 
 a. At any parallel diftance affign'd, defcribe the fquare 
 E F G H, and at the fame parallel diftance, delcribe 
 the arches I K L M, and the figures will be comple¬ 
 ted as required. 
 
 Problem VIII. 
 
 To defcribe the compound figure A BCD, luith its cir- 
 ciimfcribmg figure EKFLHMGI. 
 
 I. By problem XV. fedl. I. defcribe the geometrical 
 fquare A B C D, and by axiom IX. defcribe the femi- 
 circles a a a a. 
 
 2p 
 
 Fig. V. 
 
 Fig. VI. 
 
 Fig. VII. 
 
 Fig. VIII. 
 
 a 
 
 I 
 
 a. At 
 
Fig. X, 
 
 r. At any parallel diftance alligned, defcribc the 
 fquare E F G H; and by axiom VII. defcribc the breaks 
 K L M 1 ; and by axiom III. the arches at the an- 
 
 complete the hs 
 
 Thcfe foregoing eight inferibed figures, are very beauti¬ 
 ful forms for fountains, bafons, filh-ponds, grafs-plots, 
 or ornaments of cockle-lhells, land, borders, &c. about a 
 Itately tree of yew, holly, philerea, laurullinus, &c. or 
 llatue. Provided that you have the advantage of a ter¬ 
 race-walk, or mount, to view the fime, otherwife a plain 
 plot of grafs is fir preferable. 
 
 And to complete the idea and pradlice of fuch figures 
 in gardening, I have, for the cxercile of the young Itu- 
 dent, and txariety of choice, , for all gentlemen as deliglit 
 therein, inferred the feveral forms in figure X. which are 
 in general deferibed by the preceding rules, and may not 
 only prove a great help to invention, but alfo of ufe to 
 many gentlemen in forming fuch parts of their gardens 
 (as they relate to) in the molt elegant manner. 
 
 Thole figures marked A A, &c. arc varieties of the in- 
 terfeclions ol’ gravel, fand, and grafs walks with proper 
 centeral plots, or figures, to place ftatucs on pedelfals in, 
 as alfo the forms of the ends of parterres, or grafs-plots, 
 as circumferibe the fame. 
 
 Thofc figures marked B B, &c, are niches, or breaks 
 in hedges, walls, &c. for to place publick feats of delight 
 in, at the termination of an elegant walk, avenue, &c. 
 And, 
 
 Thofe marked D D, &c. are the forms of cabinet.s, or 
 private places of retirement, in the molt private retired 
 parts of a wildernefs, labyrinth, grove, &c. 
 
 N- B. That although hitherto I have recommended 
 thefe compound figures in gardening only ; yet the 
 ingenious ftudent in architedlure is to obferve, that they 
 are exceeding beautiful in building, as in cielings. 
 
 parrquetting, painting, paving, &c. 
 
3^ 
 
 Sect. V. 
 
 Of the Conjiruclion of the fingle, double, &c. fpiral 
 Line, Scroll, Artinatural Line, &c. for Pra^ice 
 in Gardening. 
 
 Problem I. 
 
 Plate III, 
 
 'T~'0 dejcribe 
 dijia 7 ice. 
 
 a fingle jpiral line at any ajfigned 
 
 Let it be required to defcribe the fingle fpirial line^ at Fig’ !■ 
 the dillance of the given line i k. 
 
 I. Draw a right line, as A B, and on any convenient 
 point of the fame, as at c, defcribe a circle of fuch a di¬ 
 ameter as is afiigned. 
 
 X. Take half the given line i k, and place that diftance 
 from c the center, to h and d on each fide hereof, which 
 points, h and d, are the two centers, on which the double 
 fpiral line will be defcribed. 
 
 3. Take the diftance d a, and on d, defcribe the femi- 
 circle a f. 
 
 4. Take the diftance b, f, and on h, defcribe the femi- 
 circle f h. 
 
 5. Take the diftance d h, and on d, defcribe the femi- 
 circle h g ; and fo by moving your compalTes firft to the 
 other center b, and afterwards to d, &c. you may conti¬ 
 nue the fpiral line about infinitely, which is what was re¬ 
 quired to be performed. 
 
 Problem II. 
 
 To defcribe a double Jpiral line at any dijlance af- 
 figned. 
 
 Let it be required to defcribe the double fpiral line, pig. n. 
 at the diftance of the given line m, n. 
 
 4 
 
 I. Draw 
 
^2 Of the ConjiruHion of the 
 
 I. Draw a right line, as A B, and on any convenient 
 point of the liiine, as at a, deftnbe a cij'clc of fuch a di¬ 
 ameter as is adign’d. 
 
 X. Take half the given line jn,n, and place that diftance 
 from a the center, to h and c on each fide thereof, which 
 points, b and c, are two centers, on which the double fpi- 
 ral line will be deferibed. 
 
 3. On c, With the diftance c, e, deferibe the femi- 
 circle e, s, cl. 
 
 4. On h, with the fame diftance, deferibe the femicir- 
 cle m, n, f 
 
 y. On c, with the diftance c f, deferibe the femicircle 
 /' r g, and then you will have got the double line equal ; 
 therefore, if you remove your compafles to the other 
 center b, you may thereon, with the diftance b d, deferibe 
 the femicircle d, 0, h, and on the fame center, the femi¬ 
 circle g P i, and then by removing again, firft to c, may 
 deferibe the femicircles h, u, k, and i, Q_, /, &c. as in 
 the preceding problem, which is what was to be de¬ 
 monitrated. 
 
 Problem III. 
 
 To deferibe the compound line, called the running 
 worm. 
 
 This line may be deferibed in two manners, wz. on a 
 right line, as A B, or on a fpiral line, as N K 1 G E. 
 
 I. To dejeribe the riming worm on a right line. 
 
 Pradlice. 
 
 I, Draw the right line A B, and on a, with any con¬ 
 venient opening (that will not deferibe an arch with too 
 lharp a turning to create giddinefs in walking) deferibe 
 Pig. III. jj.jg ^ pjj.,.,g opening, turn your com- 
 
 paffes from h to c, and on c, deferibe the arch b d\ and 
 then turning them d to e, deferibe the arch d /, and in 
 the like manner all the others contain’d in the line AB. 
 When this line is applied to any ufe as requires breadth, 
 as a walk through a wood, &c. that breadth may be deferi¬ 
 bed upon the fame centers, as the line it felf, and in the 
 very liune manner. 
 
 11 . To 
 
33 
 
 Jingle, double, &c. fpiral Line, &c. 
 
 II. To deferihe the running ii/or^n o-n a fpiral line, 
 
 Praftice. 
 
 I. By the preceding problem, delcribe the double fpiral 
 line F G, HI, KL, MN ; and on E a, deferibe the fe- 
 micircle, or rather arch h c, and on the fime center, the 
 arch d, e, f. 
 
 T.. With the former opening h a, turn the compalfes 
 from c to g, and on g, deferibe the arch c h, and alfo the 
 arch f i ; and with the fame openings and manner of 
 working the other arches, and their parallels, muft be de- 
 feribed. And as this running worjii is deferibed but on one ^' 2 - 
 of the two fpiral lines, therefore by giving the oihci the 
 fime parallel breadth as the running worm, and umting 
 them together at Z and F, ’twill create a variety in walk¬ 
 ing, and unexpeHedly bring out the perfon, at his place 
 of entrance, contrary to his expectation. 
 
 Problem IV. 
 
 To deferibe a treble fpiral line at anp difiance af 
 fgnd. 
 
 Let it be required to deferibe the treble fpiral line at 
 the diftance of the given line 7 n, n, 
 
 I. By problem XIII. feel. I. deferibe the equilateral 
 triangle ABC, and make tlie fides thereof each equal 
 to the given line m, n, and from the center thereof, 
 through every angle, draw the right lines B D, CD, 
 and A D. 
 
 X. On A, with the diftance A C, deferibe the arch 
 c d e, and with the fame opening on B, deferibe the 
 arch k f g\ and alfo on C, deferibe the arch B h i. (And 
 here you muft note, that in this and all other figures of 
 this kind,the feveral arches therein that compofe the whole 
 muft not be continued in one arch of a circle any farther rig. v. 
 than from one line of direftion to another, be there one, 
 two, three, four, &c. viz. in this figure; for example, no 
 arch muft be deferibed at one fweep, no farther than from 
 the line of direblion A D to the line of direblion C D, or 
 from the line of direction C D to the line of direbtion B D; 
 
 K and 
 
Of the Conjiruciion 'o^ the. \ 
 
 and the like from the line of dircflton B D to the line of 
 direction AD. 
 
 3. On A, with the diftance A i, deferibe the arch i X’ /; 
 and with the lame opening on tiie point B, dclcribc the 
 arch e ?u n ; and alio on the point C, deferibe tiic arch 
 
 4. On the point A, with the dilbmee A p, deferibe the 
 arch p, t, 11 ; and with the fame diltanee on B, defenbe 
 the arch L, q, u\ and alfo on the point C, deicribe the 
 arch n r s ; and in the like manner on the three points 
 A B and C, you may enercale the magnitude thereof, as 
 much as dclired, which is what was required to be done. 
 
 ’J'iiefe treble ipiral lines, arc exceedingly beautiful, 
 when planted with hedges of hornbeain, engliili-clm, &c. 
 and the whole environ'd with a wood, wherein may be de- 
 Icribed divers other walks (as thoi'e marked F F, &c. ) 
 that be made to unite with the tlirec feveral walks of the 
 ipiral line, as at D D D. 
 
 Thole walks marked F F, arc what I call artinatural 
 walks by reafon they are deferibed by art, and reprefent tlie 
 product of nature, which in all woods and wildenidles 
 fliould be imitated as near as polliblc, wlrich hitherto, by 
 defigners of gardens, as the late Mr. Londo7t, his followers, 
 &c. has never been thought of, or pradtifed, they always 
 obferving a ftilf heavy regular form equal in all other- 
 parts alike; fo that when any perfon had feen one quarter 
 of any of their gardens, they had then, in eftedt, feen the 
 whole, the remaining three parts being but the firft re¬ 
 peated lb many times, and thole llulf’d up with their ever¬ 
 greens at fuch a rate, that they ever had an afpedl more 
 like unto a nurfery than a pleafant garden, as intended. 
 
 The beauty of a garden (m my humble opinion) conlifts 
 in a regular, irregularity, that the parts may appear as 
 equal, and at the fame time be unequal among theinfelves, 
 and thereby, at every ftep forward, a new feene, or ffellr 
 objedt appears, and the whole becomes an everlalting en¬ 
 tertainment. 
 
 But lince this treatife is not particularly defign’d for 
 gardening, I lhall therefore forbear, and return to 
 problem V. 
 
 Problem 
 
Jingle, double, 6cc. fpiral Line, &c. 
 
 3$ 
 
 Problem V. 
 
 To defcribe a quadruple fpiral line at any dtfiance 
 aj/ign d. 
 
 Let it be required to defcribe the quadruple fpiral 
 Ime at the diftance of the given line HI. 
 
 I. By piobleni XV. fe£I. I. defcribe the geometrical 
 quaie i i 3 4., and make each lide thereof eoual to the 
 given line HI. 
 
 a. Divide each lide thereof into two equal parts^ and 
 draw the diameters, extending them infinitely; as from A 
 the center, to B C D and E. 
 
 3. On A the center, defcribe the circle e b c d, of fuch 
 diameter as fliall be affign’d. 
 
 4. The points I z 3 and 4, being the centers on which 
 
 the whole is deferibed ; therefore, on the point i, with 
 the diftance i d, defcribe the arch d a ; and with the fame 
 diftance on 1, defciibe the arch b f ; and alfo on 2 de¬ 
 fcribe the arch ^ ; and likewife on 4, defcribe the'''°'^" 
 
 arch c h. 
 
 5. Begin again, and on the point i, with the diftance 
 I h, defcribe the arch h i ; and with the fame diftance on 
 the center X, defcribe the arch al-, and alfo on 2, de¬ 
 fcribe the archand likewife on 4, defcribe the arch 
 gm-, and then begining again at the center i, with the 
 diftance i, m, &c. you may defcribe the four lines to any 
 magnitude required. 
 
 Tins kind of figuie may at laft be circumfcribed in a 
 circle (as in the figure) when ’tis applied to praftice on 
 the convexity of a mount, or concave, as that of the 
 Honourable Thoma-s Vermti?,, in the gardens of his feat 
 at Twtckenha?n Turk in the County of Middlefex, made 
 by me in the year 1721. This concave was a large fand- 
 pit, and then a perfedl nufance, and fuppofed to be inca¬ 
 pable of any improvement as would be agreeable to the 
 circumjacent parts of the gardens, then new made : but 
 when I deliver’d a draught of the fame, the former fup- 
 pofition was deftroy’d, and ’twas then demonftration fuf- 
 ficient, that inflead of its being a nufance, twould be a 
 veiy agieeable beautiful figure, as it now appears in. 
 
 And from hence it further appears, that the great ex¬ 
 pence tliat many noblemen and gentlemen have formerly 
 been put to, by the indifcreet diredlions of Mr. London, and his 
 
 emiffaries, 
 
36 
 
 Of the ConJiruBion of the 
 
 cniiflliries, in removing hills to fill up fuch concavities, to 
 make the ground level and uniform (as they in tlieu' own 
 terms call it) for the execution of their regular llufi "d 
 up parterres, flower knots, &c. with fine finakin furbelow'd 
 yews, hollies, &c. whereby the whole ever had tlic afpccl 
 of a nurlcry, more than a garden of delight, as 1 laid be¬ 
 fore ; were'not only an immenfc needlels expence, but the 
 garden it felf thereby totally ruin’d. And fince I have here 
 taken the liberty to mention that error, I will alfo enlarge 
 a fmall matter further, in relation to another, full as 
 grofs as the preceding, viz. to fully execute their regu¬ 
 lar forms, cut down many a well grown fturdy oak, elm, 
 &c. and introduces a trifling flowering Ihrub, or final! tree 
 ' of yew, holly, phylcrea, laurultinus, &c. which, in my 
 opinion, is a plain proof ol’ their ignorance ot tlie fcience, 
 as well as a crime almolt unpardonable. But to return 
 to problem ^'1. 
 
 P R O C L HM Yl. 
 
 HoJi/ to defer the an elliptical fpiral line about an el- 
 lipjis given. 
 
 Let it be required to deferibe about the ellipfis K M 
 I L, the elliptical fpiral line NOGPQ_VRST 
 W K, at the diltance of the given line X Y. 
 
 I. Deferibe the ellipfis K M I L, according to pro¬ 
 blem XXL I'ecL I. and draw the lines of direftion G F, 
 G A, H C, and H E, infinitely. 
 
 1. Divide the given line X Y into nine equal parts, 
 and take the diftance of two of thofe divifions in your 
 compalfes, and fet it on the line of direftion G F, from 
 D, the point of interlecLion, to i, and on the fame line 
 from G to a ; as alfo from B to 5, and from H to 4, on 
 the lame line. Thefe four points i, x, 3 and 4, are 
 four centers, as will deferibe the elliptical fpiral line, as 
 ibllowing. 
 
 I. On the point i, with the diftance i, N, deferibe 
 the arch N, O. 
 
 1. On 4, with the diftance 4 
 O G P. 
 
 3. On 3, with the diftance 3 
 
 p a 
 
 4. On z, with the diftance x, 
 
 av. 
 
 O, 
 
 B, 
 
 a. 
 
 deferibe the arch 
 deferibe the arch 
 deferibe the arch 
 
 y. On 
 
37 
 
 Jin^e, double^ 6cc. fpiral Line, &c. 
 
 5. On i, with the diftance i, V, defcribe the arch 
 V R. 
 
 6 . On 4, with the diltance 4, R, defcribe the arch 
 R S. 
 
 7- On 3, with the diftance 3, S, defcribe the arch 
 S T. 
 
 8. On with the diftance %, T, defcribe the arch 
 T W. 
 
 Lajily, On i, with the diftance i, W, defcribe the 
 arch W E; and in the like manner may it be defcri- 
 bed to any magnitude defired; where any perfon defires 
 to liave this line double, treble, quadruple, &c. tJiey muft 
 proceed according to the preceding rules of the foregoing 
 problems, and their defires will be anfwered. 
 
 Problem VII. 
 
 To defcribe a mint us, or fcroll, to any mag^iitude re¬ 
 quired. 
 
 As for example, Defcribe the voluta, fig. VI. with the 
 parallel diftance of its lines equal to the given line 
 X X. 
 
 1. Take the length X X, and on A, defcribe the circle 
 D 4 C 3 ; and through the center A, draw the right 
 line of diredlion S P. 
 
 2. Divide the diameter of the circle into four parts, as 
 at I, A, x; and fet off as many of thofe fmall divifions 
 on the fame line of diredlion, without the circle (as thofe 
 at 6 , 8, y, 7, &c.) as are convenient for your purpofe. 
 
 3. That being done, on the point 2, with the diftance 
 , 3, defcribe the femicircle 3 E 7 ; and on i, with the 
 
 xfame diftance, defcribe the feinicirce4 B 8. 
 
 4. On 2, with the diftance 2, 8, defcribe the arch 
 8 F G. 
 
 y. On the point 3, with the diftance 3, 7, defcribe 
 the femicircle 7 H L, and on the fame point, the femi¬ 
 circle K N P. 
 
 6 . On the point 4, and at the diftance 4 L, defcribe vni- 
 the femicircle L M O ; and on the fame point the femi¬ 
 circle G I K. 
 
 7 - On the point 6, with the diftance 6 O, defcribe the 
 femicircle O (X T, and on the fame center the femicir¬ 
 cle P R S, &c. fo that it now appears, that the oftner 
 
 L ’tis 
 
 
Of the Conftruffion, &c. 
 
 ’tis turn'd round, fo many more centers mult be added, 
 as thofe of y, 7, 8, &c. 
 
 This figure has been very much ufed in parterres, 
 flower knots, &c. but beft for an entrance into a cabinet, 
 or private place of repofe in the quarter of a wood, wil- 
 dernefs, &c. And befides all the foregoing lines of the 
 fix lall problems, there is yet another, lar liiperior to anv 
 of them, when apply'd to pradlice in rural works, and 
 is what (as I faid before) I call an artinatural line, and is 
 to be defcribed according to the following problem. 
 
 Problem VIII. 
 
 To defer lie an artinatural line, in fuch proportion rrs 
 traced hy hand. 
 
 It’s to be obferved, that there is no fet form for this 
 line, it being various according to the diferetion of the 
 hand that traces the fame ; therefore what is to be un- 
 derftood by this problem is, how to find the centers of 
 Fig. VIII. arches as will deferibe the line traced, or very near 
 
 thereunto. As for example. 
 
 Let it be required to find the centers of fuch arches 
 as will deferibe the artinatural line B C D F G H K 
 L M, &c. 
 
 I. By diferetion, divide the feveral turnings, into fuch 
 parts as doth appear to be fegments of circles. 
 
 1. In every fuch divifion make three points at plea- 
 fure ; and by prolem XI. fedt. I. find the centers thereof, 
 and deferibe the feveral fegments therein contained, and 
 they will complete the line as required. And as the on¬ 
 ly ufe that this line can be applied to, is in pleafant foli- 
 tary walks of a wood, wildernelTes, &c. therefore fuch 
 breadth as is aflign’d them, may be defcribed on the fame 
 centers parallel thereunto. See the diagram, wherein 
 one view will give more inftrudlion than many words. 
 
 Sect. 
 
39 
 
 
 Sect. VI. 
 
 Plate IV. 
 
 Of the Geometrical Contruncation of the Cube, Pa- 
 rallelepi^edon, and the folid Bodies generated 
 
 And in confideration, that the following operations 
 wholly depends upon the divilion of a right line 
 into extream and mean proportion ; I will, therefore, 
 firft lay down 
 
 H O W to divide any right line {as the given right line Fig, xv. 
 A B) in extream and mean proportion. 
 
 Pra6lice, 
 
 I. Make the geometrical fquare C D L N, with every 
 one of its fides equal to the given line A B. 
 
 X. On N, with the radius N D, defcribe the arch, or 
 quadrant, D I L. 
 
 5. Bifleft CL, CD, N L and N D, in the points O K 
 G M, and draw the diameter K M and O G. 
 
 4. Draw the right line I M, and on M, defcribe the 
 arch I F, and make B P equal to M F. 
 
 y. The diftance of B P, is the greater fegment, and 
 the point P, doth divide the line A B in extream and 
 mean proportion, as required. 
 
 Of the Contruncation of folid Bodies. 
 
 I. The folid bodies generated by the feveral ways of cut¬ 
 ting a cube, are the canted cube, the fruftum of a cube, 
 the tetraedron and its fruftum ; the o£taedron, dodecae- 
 dron, icofaedron, twelve, and thirty rhombs, of which 
 
 I the 
 
Of Geometrical Contruncation of 
 
 the tetiaedron, oftaedron^ dodecaedron^ and icofaedron, 
 (as likewife the cube) are called regular bodies, by reafon 
 
 they may be inferibed within a fphere. (See x±th book 
 of Euchd.) ■ X V ^ 
 
 z. A cube (by the 37th definition, fedl. I. part. 1 .) is 
 a Iblid body containing fix faces, each a geometrical fquare, 
 equal to each other, and every angle 90 degrees. This 
 body is very eafily made, provided every angle is an ex- 
 adh right angle, which in practice is very difficult to be 
 perform d. However, although workmen cannot be ex¬ 
 actly mathematically true, yet they come fo near to the 
 tiuth, as not to occafion any fenfible dift'erence in the 
 operation. 
 
 3. If every face of a cube be divided, as A B C D, by 
 an inlcribed geometrical fquare, as E F G H, whofe" an¬ 
 gular points divide the fides A B, B D, DC and C A, 
 each into two equal parts ; and the triangular parts, as 
 E A F, F B G, G D H, H C E, &c. are cut off, the re- 
 
 cameiCuhe. maioing body is what is called the canted cube, containing 
 14 faces, of which fix are geometrical fquares, and eight 
 equilateral triangles. 
 
 4. If within every face of a cube be inferibed an odla- 
 gon, whofe diameters are equal with thofe of the face of 
 
 Fmjium of the cube, as the odlagon F G H I K L M E, and the tri- 
 angular parts G B H, I D K, L C M, E A F, &c. are cut 
 l i". viii. remaining body^ is what is called the fruftum of 
 
 a cube, containing 14 faces, of which fix are oftagons, 
 and eight equilateral triangles. 
 
 y. Suppofe O S be i^, and O P the root of ^ Syffp, 
 and O Z the root of j divide Z Y into two equal 
 
 parts, in the point X, and draw the triangle X S O; alfo 
 draw the like on its oppofite fide, equal and oppofite there¬ 
 unto. Make O B equal to ^ of O Z, and draw B V, pa¬ 
 rallel to O S, interfedling the perpendicular X Q,, in the 
 Temedron. point R, which is the vertex of the tetraedron L M N • 
 draw the right lines V W and V T from the point V to 
 the angles T and the like from the point B, to the 
 Fia IX oppofite angles of T and W. If the triangular portions 
 V S T, B O P, G L C, &c. are cut off, the remaining bo¬ 
 dy is a prifm ; whofe fide T V W, and its oppofite, are 
 each triangles, and the other three V T P B, &c. are pa¬ 
 rallelograms. Lafly, by the points T R X, and the fide 
 of the triangle T X, drawn on the bafe, as likewife by the 
 points F R X, and the other fide of the triangle P X ; 
 divide the prifm, and thofe parts being cut off, leaves 
 
 the 
 
 40 
 
 Cube. 
 
 Fig. VII. 
 
the Cube, TarallelopipedoHj &c. 
 
 the trianglar body called a tetraedron, containing four 
 fiices, and each an equilateral triangle. 
 
 6. Divide every face of the tetraedron, as Z A, viz. di¬ 
 
 vide eveiy lide into three equal parts, and draw the 
 lines c, zgand/^; then will the figure i c i g f d, f- 
 
 be a hexagon, and if the triangular parts h a c, ih g, 
 
 J e d, &c. are cut off, the remaining body is called the 
 fiuftum of a tetiaedron, containing eight faces, of which 
 four are hexagons, and four equilateral trian-^les. 
 
 7. Siippofe a long cube, or parallelopipedon,^as S Y C F 
 
 be as follows, viz. Let S, or W X, be equal to 
 
 looqqq, and X Y, or Q^P, to the root of i- as aforefaid, .. 
 
 b,i£±9, and S X to the root of more by t thereof ® ‘ 
 11,1^, make X T, W V and O P, each equal to 
 
 one loui th of X S, 01 Z P, and draw the right line V X 
 parallel to W X, and the like on its oppofite fide * or 
 bale. ’ 
 
 Bilfea dS, in R, and draw the equilateral triano-le 
 R V T, and the fame in its oppofite fide, fo that the pomt 
 B, of the oppofite triangle, be oppofite to the point R 
 Draw the right lines V Z, V O and QY, and the like on. 
 the oppofite fide, and cut off the triangular parts TVZY 
 downwards, and its oppofita O P dS, upwards ; then will 
 there remain two equal parallelograms V T S r’ d and O 
 Z B Y. Lajily, cut off O \ R and T R above, and O V B 
 
 ""''n I ^ r'r oaae4ro„. 
 
 Will be made a folid of eight equal faces, each an equila¬ 
 teral triangle, called an odlaedron. 
 
 8 . Divide each fide of a cube into two equal narts ac, 
 
 D K by q d, H G by g z, &c. make D q, or H p &c "the 
 radius equal to loqqoo, and divide them by extream and 
 mean proportion. Then will Da, F e, &c. be equal to 
 fi,^the greater fegment, and e i, m d, 8cc to 5 8197 
 the lefler fegment. From the greatefl fegment 
 
 fide to the middle of the other, draw right lines as from 
 m to E, from 0 to z, from I to g, from e to d 8cc 
 Lajily, If you cut off each triangular prifm, as EBmr 
 log z, a e q d, 8cc. lit 1 % cuts, will be framed a folid con’ 
 taining IX equal faces, each a pentagon, called a dode- 
 caedron. 
 
 9- Divide each fide of a cube into two equal parts as 
 B C,by the right line j> e d, &c. interfering each other 
 at right angles, as the right lines l>diind a c, in the point 
 e, make &c. the radius equal to io,oo°°. and let f.v xii 
 c a, e d, and e c, be each equal to d. iso; and through 
 the points ahcd, draw the four right lines ali,})c, cd and 
 
 “ ’ d«. 
 
 II 
 
 
 
^2 . , Of Geometrical Contrmcation of 
 
 d a, extending each of them to the exteriour lines of the 
 tace^ A B, B C, C E,, and E A. Divide every face in the 
 fame proportion, and thereby is conftituted eight equilate¬ 
 ral triangles marked in the diagram 2, 3, 4, y, 6,7, 8, &c. 
 By which every angle being cut off, the body will then 
 contain fix geometrical fquares, and eight hexagons. If you 
 tco'-cviron ™‘^ke a c the bafe, the point f of the other lace lliall 
 CO,it..>0,2. vertex to cut out the triangle ac f, and //j lhall be 
 
 the bafe and / the vertex to cut out the triangle fh /, and 
 I 71, the bafe, and c the vertex to cut out tire triangle 
 Inc, and the like of all others, till e\’ery one be cut oft^ 
 and the remaining folid will be a body containing 10 
 faces, each an equilateral triangle, called an icoliicdron. 
 (t|= This body may be cut by the albrefaid lines of the 
 doiiecaedron, by drawing the parallel lines upon the cube at 
 the diltance of the leller fegment inftead of the greater. 
 
 10. Suppofe a parallelopipedon be as follows, niz the 
 length to the breadth as i is to the root of fo fliall D C 
 
 Fig,XIII. or P B be equal 10 °°°° and B A or P G, to 7,£7ro Bif- 
 ledl the lines G H, P B, and D C, in the points E L H, 
 ^ts alfo their oppofites, and draw the right lines H B, H P, 
 
 I G, I A, ID, IC, E B, E P, and their oppofites; draw the 
 diagonals D G, and F P, and their oppofites, meeting the 
 aforefaid lines of every angle, and thereby conllitute tri- 
 
 is Rhotnbt. angles, fuch as D I G, &c. Laftly cut away the angle P, 
 by the triangle DIG, and the like of others, and thereby, 
 at eight fuch operations, will be left a folid body, containing 
 
 II faces, each a rhombus, called the body of 12 rhombs. 
 
 11. Divide every fide of a cube by extream and mean pro¬ 
 portion, as the fides ad, d g, g k, and k a-, in the points 
 h, c, e, f, h, i, I, n, where each fide is equal to io °oo°. 
 and tlie leller fegments ak, c d, d e, fg, g h, i k, k 1. 
 
 Fig. xiv. and « a, eacli equal to 3 
 
 Draw right lines from the terms of the lelfer fegments, 
 on the one fide, to the greater on the other fide, as the 
 lines n c. Id, e k, and f i, which will be parallel to each 
 other. Alfo interfedl them with the like parallels, as h e, 
 a f n g, I h, and draw the right lines eg, h k, h d, i a. 
 Divide every fiice of the cube in the fime manner, and 
 -0 Khombt then will the cube be prepared for the operation. About 
 every lolid angle of the cube are three triangles, as the 
 triangles i, 2, and 3, about the angle a, and the tri¬ 
 angles 4, T, 6 , about the angle d, &c. Therefore every 
 angle muft be cut three times, always obferving to con¬ 
 tinue each line, as a part is cut away; otherwife 'twill be a 
 confuled work; and thereby at 24 fuch operations, will 
 
 3 appear 
 


 r 
 
the Cube, Parallelopipedon See. 
 
 appear a folid body, containing 30 faces, each a rhombus, 
 and is called the 30 rhombs. 
 
 Thefe bodies are not only very beautiful in divers parts 
 of building, but alfo in gardening, being placed on a pro¬ 
 per pedeftal, with a fun-dial delineated upon every face, 
 which may be fo contrived as not only to fhew the hour 
 of the Day, in all parts of the world, according to the 
 feveral accounts of time; but alfo all the moft ufeful parts 
 of aftronomy, as the fun's place, declination, amplitude, 
 light afccnlion, altitude, azimuth, riling, letting, length 
 ol day and night, beginning and ending of twilight, lequa- 
 tion of time, &c. 
 
 PART 
 
THE 
 
 P R A C 1 I C I' 
 
 O F 
 
 Avcbitediire, Gardening, Menfiiration, and 
 
 Land-Surveying, Geometrically demonllrated. 
 
 PART II. 
 
 I. The Geometrical ConjiruBion of the Tvfcan, Do- 
 rich, Jonick, Corinthian Compojita, Trench and 
 Spamjfh orders of Architecture, according to any 
 proportions ajfigned, as alfo of all kinds of plans and 
 uprights whatfoever. 
 
 II. The Geometrical and Trigonometrical Conjiruc- 
 tion of all forts oj Plans, or Draughts of Gar¬ 
 dens, TLilderneffes, Labyrinths, Groves, &c.' 
 and Maps of Cities, Towns, Tarifjes, Lord- 
 Jhips, EJiates, Farms, &c. 
 
 Sect. I. 
 
 Of the Geometrical Confiruhiion of Flans and Uprights. 
 
 P L A T F. V. 
 
 'T^O delineate the geometrical plain, or ichnography of 
 a building is to accurately defcribe a geometrical 
 figure of the feveral parti thereof in true proportion. 
 
 The 
 
Of the GeotHetrical ConfiruBion 
 
 The common meafure ufed herein is the englifh foot, 
 divided into ix equal parts, called inches, each being e- 
 qual to the length of 3 barley corns placed in a right line, 
 therefore, an englifh foot is equal to the length of 5 d bar 
 ley corns. The inches graduated on a foot, or two foot 
 rule, are fubdivided in 4, 8, lo, ix, &c. equal parts, 
 according to the pleafure of the architedl, &c. 
 
 The length, breadth, depth, &c. of any building, (or 
 its parts) are called its dimenfions, and the meafuring of 
 thofe dimenfions, is called taking the dimenfions. 
 
 All dimenfions, or meafures of feet and inches, when 
 taken, are thus written and exprefled, viz. a dimenfion, 
 whofe length is fix feet and ten inches, is written 06f. loi. 
 and fixty two feet, and five inches, thus dx/i yi. alfo 
 if a dimenfion be fourteen feet and eleven inches in length, 
 by nine feet fevcn inches in breadth, and two feet ten 
 inches in depth, or thicknefs, ’tis thus written, 
 
 f z. 
 
 14 : II. 
 
 09 : 07. B>&Ci 
 
 ox : 10, Dj 
 
 To exprefs one, two, three, &g. feet by a plain fcale (or 
 fcale of equal parts); every fuch equal part (as an inch, &c.) 
 doth reprefent a foot, and two inches, two feet, &:c. and if 
 the inches are divided into i x equal parts, each of thofe 
 parts will reprefent an inch. Therefore fix feet and ten 
 inches, is rcprefented by fix inches and f-°, and fixty two 
 feet five inches, by dx inches y parts. And what is here 
 faid of the divifion of an inch into ix equal parts (for the 
 rcprefentation of inches) the fame is to be underflood in 
 the divifion of any other length, as j, (■, c, &c. of 
 an inch, foot, yard, &c. 
 
 When the dimenfions of a building are taken in foot mea¬ 
 fure only (without regard had to inches, which in fome 
 works is very common) then any equal divifion, as ismoft 
 convenient, may reprefent one foot, as of an inch, 
 which before reprefented but one inch, may now reprefent 
 one foot, and confequently an inch ix feet; and the like 
 of any other equal part, or divifion whatfoever. And for 
 the better information hereof, that the young Undent 
 may have a perfedl clear idea, I will here demonflrate the 
 conflrudlion of fuch plain fcales, as is mofl convenient for 
 his purpofe. 
 
 N 
 
 Problem 
 
Of the Geometrical ConJiruBion of 
 
 Problem I. 
 
 To make divers fcales of equal parts, as fjall reprefent 
 feet and inches. 
 
 I. Draw the right line D B, and at B erefl; the per¬ 
 pendicular B A, and make B A equal to B D. 
 
 Divide D B into iz equal parts, at the points i z 3 
 4, &c. And alfo A B at the points i z 3 4, &c. 
 
 3. Draw, or continue B D to fuch a lengtJi as you would 
 have the fcale to contain ; as to E, and draw A A paral¬ 
 lel, and equal in length thereunto. 
 
 4. Draw AE, and divide it into iz equal parts, at the 
 points 1 z 3 4, &c. 
 
 r. Draw the lines i, i. z, z. 3, 3. 4, 4, &c. parallel 
 to A A and E B. 
 
 6 . From the point A to D, draw the right lines A D, 
 A I, A z, A 3, A 4, A y, A d, A 7, A 8, A 9, A 10, A 
 11, and the line A B is the i z divilion. 
 
 The IZ centeral lines thus drawn, do divide the ends 
 of the IZ parallels, each into iz equal parts ; therefore 
 each of thofe lines, fo divided, doth reprefent one foot 
 divided into iz inches, as Z i, and if you take the dif- 
 tance Z i in your compafTes, and fet that diftance from Z 
 to F', and from F to G, and from G to H, &c. each of 
 thofe divilions fliall be a foot, and equal to Z r, the foot 
 divided in iz parts. And to take off with vour compalTes 
 any number of feet and inches required, proceed as 
 follows. 
 
 Let it le required to take off four foot eleven inches. 
 
 Pradlice. 
 
 Set one point of your compalTes in the point 4 I, and 
 extend the other on the fame line, to the point of inter- 
 feftion of the centeral line A 11, and the line i, i, from 
 which you take the meafure, and that length lhall truly 
 reprefent four foot and eleven inches, according to the di¬ 
 vilion of that line. And what is here faid in refpedl to 
 the divilion of this line, the fame is alfo to be underllood 
 of all others. And from hence it appears, that therein 
 
 there 
 
Platts and Uprights', 
 
 t^here is contain’d 12 various fcales, and each reprefentine 
 feet and inches, which is what was required to be done 
 The conftruftion of the fcales of foot meafure fig. II. are Fi? ii 
 made by the very fame rule, only the fides K L and L ‘ 
 
 M, are divided into 10 equal parts each, inftead of into 
 11, as in fig. I. 
 
 r c now proceed to the conftruaion of one other 
 ufefeil fcale, which is called a fcale, or line of chords, and 
 is of as great ufe in meafuring the angles of a building, as 
 the other in meafuring the fides, &c. ° 
 
 47 
 
 Problem II. 
 
 lem°th ^ chords, to any aj/igrid 
 
 Definitions. 
 
 A chord, or lubtenfe, is a right line joining the extre- 
 AM C ’ fo A C is the chord of the arch 
 
 A line of chords is no other than 90 degrees of the 
 
 arch of any circle, transferr’d from the limb of a circle to 
 a right line. 
 
 Every circle (great or fmall, fee problem IIL part I. 
 lett. 1.) IS divided into 3 do equal parts, term’d degrees,F:g-W- 
 
 fntnoT'^^ a femicircle into 180, and a quadrant 
 
 into 90. The lemidiameter of a circle, or the fide of a 
 quadrant, is always called the radius, and is ever, in all 
 cades, equal to do degrees of the fame; therefore when 
 the vvord radius is nereafter mention’d, then fixtv degrees 
 is to be underllood alfo. ^ ^ 
 
 Conftrudion. 
 
 1. Defcribe the femicircle B M D O, and on O erect 
 the perpendicular O M, which will divide the femicircle 
 into two quadrants. 
 
 X. By problem XXX. part I. fed. I. divide the arch M 
 
 D into 90 equal parts or degrees. 
 
 3. On the point D fet one foot of your compafles 
 and extend the other to 10, and defcribe the arch 10 10 
 then open them to 20, and on the fame point D defcribe 
 tne arch 20, 20, and in the fame manner the arches ?o, 
 3 30, 
 
48 
 
 of the Geometrical Conjlruhlion of 
 
 30, 40, 40, 5-0, 5-0, 60, 60, 70, 70, 80, 80, and 90, 
 90, which feveral arches will interfedl the diameter B D, 
 in the points 90, 70, do, 5-0, 40, 30, ao, and 10, and 
 divide it into unequal parts. This line, thus divided, is 
 the line of chords, divided to every tenth degree, and by 
 tlie llune rule you may divide it to every degree, and 
 therefore needs no further explication. And as the only 
 life of this line is to meafure the quantity of any angle, 
 therefore ’twill not be improper, hrft to demonftrate the 
 variety of angles. 
 
 Demonftration. 
 
 When two right lines, as E F and F G, join each other, 
 in a right lined pofition, they then make no angle, but do 
 conftitute a right line equal to both their lengths; fo the 
 line E F and F G, meeting together in a right line pofiti¬ 
 on, at the point F, do conftitute the right line F G. But 
 when tw'o right lines meet, and not in a right lined pofiti¬ 
 on, as the right lines A D, and H D, (or A D and B D, or 
 H D and D C) fuch lines, by fuch meeting, form an an¬ 
 gle. The meeting of fuch lines may happen in three feve¬ 
 ral politions. 
 
 I. Tw'o right lines may meet as the right line B D on 
 the line A C, in the point D, making the diftance from B 
 to A, equal to B C, 'viz. the line B D, perpendicular to 
 the line A C, and thereby conftitute two equal angles, each 
 containing a quadrant or arch of 90 degrees, and are called 
 by the name of right angles. Therefore whenever a right 
 angle is mention’d, an angle of 90 degrees is to be undcr- 
 ftood. 
 
 1. Two right lines may meet as the right lines A D and 
 H D, and thereby conftitute an angle, lei's than 90, and 
 therefore is called an acute angle. 
 
 3. Two right lines may meet, at the right lines FI D and 
 D C, and thereby conftitute an angle, more than 90 de¬ 
 grees, and therefore is called an obtufe angle, and the fum 
 of all is, that an angle is either acute, right, or obtufe. 
 
 An acute angle is that whofe meafure is lefs than a qua¬ 
 drant, or arch of 90 degrees. 
 
 A right angle is that whofe meafure is a quadrant, or 
 arch ol' 90 degrees. And, 
 
 An obtufe angle is that whofe meafure is more than a 
 quadrant, or arch of 90 degrees. 
 
 The meafure of an angle is an arch of a circle, deferibed 
 upon the angular point, intercepted betw'een the two fides, 
 
 as 
 
 Fig. IV, 
 
49 
 
 ^lans and Uprights. 
 
 as containeth the angle, (an angle is always exprelTed by 
 three letters, whereof the middle letter always denotes the 
 angular point, as for example, if you expreYs the angle 
 Z X Y, the letter X fignifies the angular point, and the like 
 of all other angles, in general j. 
 
 The complement of an angle, (or arch) is fo much of an 
 arch, as the arch that meafures the angle wanteth of a qua¬ 
 drant or arch of ninety degrees. So if an angle containeth 
 6^0 deg. the complement to an arch of qo deg. or quadrant 
 is 30 deg. and the like of any other angle. 
 
 All angles concurring upon one right line in a center, 
 being taken together, are equal to a femicircle, or 180 de- Fig- v. 
 grees. 
 
 So the angles of the right lines a a a, &c. meeting at 
 the point C, are (taken together) equal to a femicircle or 
 180 degrees. 
 
 Having thus fliewn the conftruaion of plain fcales, 
 fcales of chords, &c. and the nature of angles, I lhal! now 
 proceed to apply them to praHice, in the delineating of 
 plans in general. 
 
 Problem HI. 
 
 T0 make a plan equal to a plan ghen. 
 
 Let it be required to make the plan X Y, equal to the 
 given plan T V. 
 
 I. By problem VII. part I. fed. I. having firft drawn 
 the line i, a, and made the fame equal to A B, make the 
 angle i, i, 3, equal to the angle BAG, 
 
 X. Make the line i, 3, equal to A C, and make the 
 angle i, 3, 4, equal to the angle A CD. 
 
 3. Make 3, 4, equal to C D, and make the angle 3, 4., 
 S, equal to C D E. 
 
 4. Make 4, j, equal to D E, and make the angle 4, 
 S, 6, equal to DEE, and by the fame rule pafs through 
 the whole, and thereby you will complete the plan X Y, 
 which will be equal to the given plan T V. 
 
 Problem IV. 
 
 fecond example. 
 
 Let it be required to make the plan T Z, equal to the 
 given plan W X, 
 
 O 
 
 I. Make 
 
50 
 
 Fig. VII. 
 
 Of the Geometrical Conjiru^ion of 
 
 I. Make the parallelogram i, i, 9, 10, equal to A B 
 L and draw the diameters ag, ag, and xr, xi, 
 
 X. Make i, g, x, 4^ 7, 9, and 8, lOj equal to A G 
 B H, I I., and K M. 
 
 g. Make 19^ and xo, x, equal to A E and 
 F B. 
 
 4, Continue the longeft diameter infinitely, and make 
 xg, id, equal to VV T, and by problem XI. febl. 1. parti 
 deicribe the arch 19, id, xo. 
 
 5. Make x.i, y, and xx, d, equal to OR and O S, and, 
 by tlie albrefaid problem, defcribe the arches g, y, 7, and 
 4, d, 8. 
 
 d. Continue the end 9, 10, and make 11, 9, and 10 
 IX, equal to N L and M O. " ’ 
 
 ' 3 . 14, equal to 
 
 ^ P 1 ^nd make ig, 17, and 18, 14, equal to P G 
 and D CL- 
 
 8. Continue ig, xg, infinitely towards ly, and make 
 xg, ly, equal to VV, V. 
 
 9. By problem XL parti, defcribe the arch 17, ly, 
 18, and 'twill complete the plan as required. 
 
 N. B. If any plan has a thicknefs, as the walls of a 
 budding, &.C. that thicknefs (be what it will) mult be 
 drawn parallel to the external figure, in fiich iiropor- 
 tion as the thicknefs is found. 
 
 Problem V. 
 
 To meafure {or take) the quantity of an ande ly the 
 help of a two foot rule, five foot, or ten foot rod 
 
 07ily. 
 
 Fig. Vlll. 
 
 Let C A B be the angle of a building, and 'tis required 
 to draw upon paper an angle equal thereunto. 
 
 1. Fiom A. towards B, mcafure, or let off, any number 
 of feet (as for mflance in this example five foot) and al- 
 fo from A towards C, at the points y and y. 
 
 2. Meafure the diftance between y and y, and note it 
 down on paper. 
 
 g. To draw the fame upon paper, firfl draw a line at 
 pkafure, as D E, and from any fcale of equal parts take 
 oft five parts, reprefenting the five foot fet oft' from A, 
 3 the 
 
Plans and Uprights. 
 
 the angle aforefaid. With this diftanee fet one foot onD^ Fig.vin. 
 and with the other defcribe the arch o, o. Take ten foot 
 in your compafles (the diftance between s and y) and fet 
 one foot in o, and with the other interfeiT the arch o, o, 
 in the point P, through which^ from D, draw the right 
 line D P N ; fo lliall the right lines D E and D N,form 
 the angle NDE, which lliall be equal to the angle CAB, 
 as required. And what is here faid concerning the 
 taking off this angle, the fame rule is allb to be under- 
 ftood of all other angles in general, be tliey acute, right, 
 or obtufe. 
 
 Problem VI. 
 
 Howto take the plan of a crooked line, or wall, which is 
 not any part of an ellipfis or circle. 
 
 Let it be required to defcribe the plan of the crooked 
 line ABC. 
 
 Pradlice. 
 
 I. On a piece of paper defcribe a crooked line, as near 
 like the crooked line A B C as you can, and draw the 
 ftreight line A C ; this being done, meafure two foot (or 
 more according to tlie nature of the curve) from C in a 
 right line towards A, as from C to x. 
 
 a. Meafure from x to the crooked line, as to e, and on 
 your paper, or eye-draught, make a mark reprefenting the 
 point X, and from thence draw a line to the curve, to 
 reprefent the offset x e, and thereon let down the meafure 
 of the offset x e. 
 
 3. At a proper diftance from x, as at 4, take another 
 offset, and fignify the fame in your eye-draught with the p; 
 true meafure of the fame; as alfo its diftance from C, and 
 in the fame manner proceed, making as many offsets as 
 the turn of the curve requires, till you have taken the 
 whole down. This being done you may defcribe the fame 
 on paper, truly thus : 
 
 I. Draw the line A C, by a fcale of equal parts, equal 
 in length to A C. 
 
 X. Set from C to x, the diftance meafured, and on 1 
 ereff the perpendicular x e, and thereon fet oft' the length 
 of that oftset, as fpecify’d in your eye-draught. 
 
 3. Set 
 
Of the Geometrical Confiruclion of 
 
 _ Set the diftance C 4, and on 4 eredfc the perpendicular 
 4 f, and thereon fet off the length of that offset as mea- 
 fured. And in^ the like manner lay down the diftance 
 of every offset from one another, and their proper lengths, 
 and then you have the ends of all your offsets, through 
 whicli you may exadlly trace the crooked line, as re¬ 
 quired. 
 
 Note, That the greater the number of offsets are ta¬ 
 ken, the more exad the curve may be drawn. 
 
 Problem VII. 
 
 Hoiv to take the plati of any huilding ivhatfoever. 
 
 The firft ftep to this performance, is to make an eye- 
 drauglit of the fame, viz. a rough draught drawn by 
 hand only, exprefling every wall, partition, room, door, 
 chimney, window, &c. and the larger thefe kinds of 
 diaugjits aie made, the better tis for you, by reafon vou 
 have good room to fet down every dimenfion, which ni a 
 hnall draught cannot be done. 
 
 I.et it be required to make a plan of E F G H, which 
 is fuppofed to be a real houfe. 
 
 Pradicc. 
 
 r. Make your eye-draught thereof as A B C D, and 
 therein reprefent every door, window, paflage, ftair-cafe, 
 partition, thicknefs of walls, rooms, &c. 
 
 a. With your five foot, or ten foot rod, meafure the 
 length and depth withoutfide, and note thole meafures 
 down to each refpedive fide, or length. 
 
 5. Meafure the thicknefs of thofe outfide walls, and 
 note them down alfo. 
 
 4. By problem XXXII. fed. I. part I. examine every 
 angle, whether they be fquare or not. If they are found 
 to be fquare, note it down, and if not fquare, as acute, or 
 obtufe, then meafure the quantity of one by problem V. 
 hereof; and thereby, with the length of the four fides 
 ^iven, you may, when you come to draw the plan of the 
 lame, by problem XIX. fed. I. part I. delineate the fame 
 exactly. 
 
 4 y. Meafure 
 
53 
 
 'Plans and Uprights, 
 
 S. Meafure the exa£l breadth of every door and win¬ 
 dow withoutlide^ and alfo the peers of brickwork between 
 then!;, and fet thofe meafures down to each refpedlive 
 part. The outfide walls being thus nieafured, the next 
 proceeding to be made is in the diftribution of the parts 
 of the houfe ; therefore walk over the fame, and as you 
 walk draw every particular room, with its chimney, doors, 
 
 &c. as near the truth as may be, as alfo every ftair-cafe, 
 paflage, clofet, &c. which being finillied, your eye-draught 
 is now fitly prepared to receive every dimenfion that is to 
 be taken. To which proceed, JirJi, as ’tis bell to begin in a 
 corner room. Therefore make a begining at I, where you Kg- x. 
 mull meafure the exadt length of every part thereof, as 
 alfo the thicknefs of its party walls, or partitions, and 
 note each meafure down feverally in its refpedlive place, 
 and then proceed to K, and there perform the fame, as 
 alfo at L, M M, N, O, P, Q_, &c. and thereby you’ll 
 have taken the juft dimenfion of every part contain’d on 
 that floor. And in the very fame manner, may you take 
 the plan of the cellars and other lower offices, or cham¬ 
 bers, when required. 
 
 Tour eye-drau^dt leing this finiJ}jed, the next work is to 
 delineate a true draught thereof from thofe meafures 
 taken, which thus perform. 
 
 I.'Ey the meafures taken, it appears the houfe is a paral¬ 
 lelogram do foot in front, and 40 foot in depth; there¬ 
 fore, with your fcale of equal parts, deferibe a ( arallelo- 
 gram, whole longeft fides are each equal to 60 parts, and 
 the fhorteft to 40 parts. 
 
 z. The thicknefs of the outfide walls are found to be 
 three bricks in thicknefs, which is equal to two foot and 
 three inches, therefore, at the diftance of two foot and 
 three inches, of your fcale, draw the interiour line, parallel 
 to the exteriour, and thofe two parallel lines do reprefent 
 the thicknefs of the outfide walls. 
 
 3. By the meafures of the eye-draught the diftance from 
 the angles to either of the adjacent windows is four foot, 
 as alfo every window and peer of brickwork between. There¬ 
 fore, divide the external lines A B, B D, A C and C D, 
 in fuch proportion, as the eye-draught doth exhibit, as 
 alfo the internal line likewife, and thereby every window 
 and out doors are truly divided in their proper places. 
 
 P 
 
 4. Draw 
 
54 
 
 Of the Geometrical Conftruhlion of 
 
 4. Draw the diameters O Iv and M M, and on eadi 
 fide the diameter OK fet of j-the breadth of the halls O and 
 K, viz. 8 foot 10 inches^ and draw on each fide the lines 
 V V and V V, and alfo the thicknefs of tliofe walls^ as they 
 are found to contain. 
 
 f. On each fide the diameter M M fet oft' two foot 
 the { breadth of tire entrance, and draw the parallel lines 
 X X and X X, wdtich will divide the parts N L and P I 
 into four equal parts. ’ 
 
 6 . Draw the thicknefs of the lines X X and X X as 
 they arc found to contain. 
 
 7. Give to the door of every room, as Z Z Z Z Z its 
 proper breadth, and from thence fet off the fide of each 
 
 Fig. X. room towards the chimney, and draw the front of every 
 chimney, as alfo fet off' the jaumes and chimney likewife, 
 according to every refpeftive meafure of your eye- 
 draught. 
 
 Laftly, Divide tlie two ftair-cafes according to each ref- 
 peftive meafure, and the plan will be completed, as re¬ 
 quired. 
 
 N. B That the fpace contain’d between any two pa¬ 
 rallel lines, that reprefents the thicknefs of a wall 
 muff; always be fill’d up with Indian ink, &c. that 
 thereby the fame may be underftood to be a Iblid, as 
 likewife thebafisof columns, as^'/// and /, &c. and 
 thofe parts that reprefent a door, or window, to be 
 left clear without any tilling up. See fig. X. 
 
 io- I do advife the young pracTitioner to conlidcr this 
 problem well, and to pradlicc herein for fome time, 
 before he proceeds any further, that he may be per¬ 
 fect, which may be done by a few days practice. 
 
 This problem of taking the plans of houfes, is one of the 
 mofl: ufeful in architediure, and the ealielt to be acqui¬ 
 red ; therefore confider the reafons of the lame judici- 
 oufly before you proceed to problem VIII. 
 
 Problem 
 
Vlans and Uprights. 
 
 55 
 
 Problem VIIL 
 
 How to draw the geometrical upright {or front) of any 
 luilding. 
 
 Let it be required to draw a geometrical upright of the 
 houfe A C B Dj which is an elevation raifed from 
 the plan E F G fig. X. 
 
 I. Make your eye-draught X, and then repair to the 
 building, and meafure the whole front from B to D, which 
 being juft; 6 o feet, write down the fame at the bottom of 
 your eye draught. 
 
 X. Meafure the wliole height from the ground at B to 
 A, which being juft 37 feet, write down the fame on your 
 eye-draught againft the middle of the height. 
 
 3. Meafure the diftance from B to 0, from 0 to p, from 
 p to q, fi-om q to a, from ti to r, from r to r, from s to u, 
 from u to w, from w to x, from x to y, from / to ;k, from 
 zto za, from z a to zh, and from zb to D; and write 
 down the feveral mealiires in each refpeftive place. 
 
 4. Meafil) e ihc diftance from G to h, from h to i, from 
 z to h, fi om k to /, fi om / to m^ fi'oni m to n^ and from 
 n to A, and write down the feveral dimenfions, or mea- 
 fures, in their refpeftive places, as may be feen in the 
 eye-draught. 
 
 The meafures, or dimenfions, being thus taken, and 
 noted in your eye-draught, proceed to the delineation 
 thereof as follows. 
 
 I. Make the parallelogram A CBD, in fuch proportion 
 that A C and B D, do contain do feet of any plain fcale 
 and the fides A B and C D 37 feet, as noted in the 
 eye-draught. 
 
 X. On the lines B D and A C, fet off the feveral mea- 
 
 fines h, 0 , p, q, u, r, s, t, u, w, x, y, z, z a and 
 
 ::: b. 
 
 3. Draw the lines 0 0, p p, q q, u u, r r, ss, tt, u u, 
 
 ww, X X, yy, z z, z a z a and zb zb. 
 
 4. On the lines B A and D C, let off the feveral mea¬ 
 fures 3, 4, 8, 4, 8, 4, d, at the points h, i, k, /, m, n, 
 E, and di aw the lines h h, z z, k k, 11, mm and n n, 
 which will mterfedb the former, and truly form every 
 Window, door, &c. contained therein, and thereby com¬ 
 plete the geometrical upright as required. 
 
 Fig. XL 
 
 3 
 
 Problem 
 
Of the Geometrical Confiru^ion of 
 
 Problem IX. 
 
 Plate VI. 
 
 To delineate the geometrical upright of any of the 
 five orders of architecture {contained in any Jiructure) 
 according to any proportion affiigned. 
 
 For Example, 
 
 Let it be required to delineate the geometrical upright 
 of the attick bafe, with the dorick capital, architrave, 
 freize and cornilh. 
 
 The meafuring rod, with which the feveral parts of a 
 column and its entablature are meafured, is the diameter 
 of the column divided into do equal parts, called mi¬ 
 nutes. Every architedl divides the members, or parts of 
 his orders, in fuch proportion as he thinks molt agreeable, 
 as may be feen in the lall folding pages hereof^ wherein 
 are exhibited, not only the geometrical profiles and 
 fedtions of the moft noble antient orders of the Romans, 
 but alfo of Vitruvius, Talladio, Scamozzi, Serlio, Vig¬ 
 nola, 2 ). Barharo, Cataneo, L. B. Alberti, Viola, Bullant, 
 T. Be Lorine, Terrault, Le Ckrc, A. Bojfe and Michae’l 
 Angelo ; which I thought fit to fubjoin to this work, in 
 fuch a manner, as for the young ftudent to behold, at one 
 view, the great variety contained among them, as well 
 as to make choice of fuch as might bell fuit his pur- 
 pofe. 
 
 The divifion of each member is a line, and the dif- 
 tance between any two of thofe lines is called the height 
 of the member, as the diftance between the right lines 
 A A and B 40, viz. the line A B, or A 40. 
 
 The projedlure of every member is that length con¬ 
 tained between the centeral line of the column and the 
 termination thereof; the entablature of any order is the 
 architrave, freize, and corniflr taken together. 
 
 Operation. 
 
 I. Let X X be equal to the diameter of a given co¬ 
 lumn, divided into 60 equal parts or minutes, by the 
 help of which well deferibe the attick bafe, as required. 
 
 And 
 
57 
 
 Plans and Uprights. 
 
 And as^’tis ufual for all architedls to prefix to every mem¬ 
 ber its exadt height and projedlure as in the feveral figures 
 XVI, XVII and XVIII, therefore draw the right line A B, 
 and make it equal to 40 min. (as there written). 
 
 а. On A, erect the perpendicular A D, and let it repre- 
 fent the centeral line of the column continued through the 
 bafe; alfo eredl the perpendicular A 40, and continue it 
 infinitely. 
 
 3. Becaufe the height of A B and A 40 is 10 min. 
 therefore fet off 10 min. from A to B, and from A to 40, 
 and draw the line B 40. 
 
 4. The height of the next .member Be, is 7 min. there¬ 
 fore fet off 7 min. from B to c, and draw c K parallel to 
 B 40. 
 
 S- The next member C E is i min. - in height; there¬ 
 fore fet oft from c to E one min. jjand draw the line E L 
 infinitely, and parallel to C K. 
 
 б. Becaufe the lines CK and E L, are each 36’ min. j in 
 length ; therefore fet off 3d min. ~ from CtoK, and from 
 E to L, and draw the line L K. 
 
 7. Continue L K to M, and divide K M into two e- 
 qual partEi' at N, and thereon deferibe the arch M P K. 
 
 8. The next member E F, is 4 min. ^ in height; there¬ 
 fore fet off 4 min. from E to F, and draw the line F n, 
 infinitely. 
 
 9. The next member F G is i min. ~ high; therefore fet 
 off I mill- ~, and draw the line G 0, infinitely, and paral¬ 
 lel to all the former. 
 
 10. Becaufe the lines F n, and G 0, are each 3 y min. 
 in length ; therefore fet oft 35- min. from F to n, and 
 from G to 0, and draw the line n 0. 
 
 11. Draw the line n L, and divide it into two equal 
 parts in 7 n, and thereon, with the diftance m n, deferibe 
 the arch « Q_ L. 
 
 12. The next member GH is five min. ~ high; there¬ 
 fore fet up y min. from G to H, and draw the line H ^ 
 parallel to G 0, and extend it infinitely. 
 
 13. The next and laft member is one min. ^ in height, 
 therefore fet up one min. ^ from H to I, and draw the 
 line I r parallel to the former, and extend it infinitely alfo. 
 
 14 Becaufe the lines H q and I r, are each equal to 33 
 min. (• ; therefore make H q and I r, each equal to 33^' 
 min. and draw the line q r. 
 
 ly. Draw the line q 0, and divide it into two equal 
 parts at P, and thereon, with the diftance P 0, deferibe 
 the arch 0, R, q, 
 
 0, 
 
 g. XVI. 
 
 16 . Make 
 
^8 Of the Geometrical Conjiruhlion of 
 
 16. Make H S equal to 30 min. and on the point S e- 
 recl the perpendicular S t, and make S t equal to twice 
 S q. 
 
 17. Draw the right line t r, and on r, with the diftance 
 r t, defenbe the arcli t u, and with the fame opening on t, 
 the arch r u, interfering the former in n. 
 
 18. On the point u, deferibe the curve r t, and ’twill 
 complete one half of the attick bafe and bafe of thelhait, 
 as required. 
 
 -2.. ]^et it he required to delineate the geimetrical upriqljt 
 apnai. ^jr dorkh cupkal, fig. XVII. 
 
 1. Draw the right line A a, infinitely, and at A crefl: 
 the perpendicular A I, for the centeral line of the capital. 
 
 2. At I min. - dillance from A a, draw the line E b pa¬ 
 rallel to A a, and make A a and B h each equal to a8 min. 
 and draw the line a b. 
 
 g. At 3 min. ~ diftance from V,h, draw the line Cd in¬ 
 finitely, and parallel to B b. 
 
 4. Continue ah to and divide b e into two equal 
 parts in the point c, and thereon, with the diftance c h, 
 deferibe the femicirclc b 30, f. 
 
 y. Set up 9 min. from C to D, and draw the line D f, 
 infinitely. 
 
 6. Take 1.6 min. in your compallcs, and fet that dif- 
 tance from C to d, and from f) to f\ and draw the line 
 
 d f. 
 
 7. Set up 3 min. 7 from f)to E, and draw E L infinite¬ 
 ly- 
 
 8. Divide E D into three equal parts at tire points a a, 
 and draw the line a g and a h, infinitely. 
 
 9. Set 30 min. from E to h, and continue d f to L, 
 and divide L k into three equal parts at the points m and 
 n, from which draw lines parallel to f L, and they lliall 
 terminate the lines f g h i. 
 
 10. Set up 6 min. and ^ from E to F, and draw the 
 line F n infinitely, and parallel to the line E L. 
 
 11. Make F n equal to 36 min. and draw the line K n, 
 which divide into y equal parts, and on the points K and 
 n, with an opening of 4of thofe divifions, deferibe the ar¬ 
 ches X X and 4 4, interfefting each other in the point m, 
 whereon, with the radius m K, deferibe the arch K fi. 
 
 IX. Set up 6 min. from F to G, and draw the line 
 G 0, infinitely, and parallel to the line F n, and at n e- 
 re£l the perpendicular n 0, and make G 0 equal to 37 
 min. 
 
 13. Set 
 
59 
 
 Vlans and Uprights. 
 
 i;. Set up X mill, from G to H, and draw the line 
 H j infinitely, and parallel to G o, and make H ^ equal 
 to 3 9 min. as alfo the line 1 1, at the parallel diftance of 
 I min. J-. 
 
 14. Fig. Z reprefents the face of the member Hi 0 G, tig.xvii. 
 which defcribe as follows, mz. draw the line 0 S, and 
 billeet it in R, and divide each half into 7 equal parts, and 
 with the diftance of 6 of thole parts, on the point 0, de¬ 
 fcribe the arch 7 7; alfo with the fmie diftance on R, de¬ 
 fcribe the arch 6 6, interfering the former in the point 8, 
 and alfo defcribe the arch y y. This being done with the 
 fame opening on the point S, defcribe the arch 3 3, inter¬ 
 fering the laft in the point 9. 
 
 ly. The points 8 and 9 are the centers of the arches 
 O R and R S, which compofe the face of the member, as 
 required. 
 
 16. Fig. N reprefents the fillet Bh a A (under the af- 
 tragal C d e i B) with a ferion of the fliaft, which defcribe 
 as follows, vix. bilfer n A in i, and make n m equal 
 to three times n i, and draw the line A m, and on m, with 
 the diftance m A, defcribe the arch A S, and on A the 
 arch m t, interfering the former in r, which is the cen¬ 
 ter of the arch or hollow A m, as will complete the ca¬ 
 pital with the aftragal as required. 
 
 3. Let it he required to delineate the geometrical up-^onck.Ar- 
 right of the dorick, architrave, freize and cor7iice, fig. 
 
 XVIII. Cornice. 
 
 Prarice. 
 
 I. Draw the line A a, at pleafure, and at a erer the 
 perpendicular a, 0, which is to reprefent a continuation of 
 the centeral line, from which every meafure of projerure, 
 and on which every meafure of height is to be accounted! 
 
 X. At the parallel diftance of 11 min. draw B c infi¬ 
 nitely, and make a A and B h, each equal to %6 min. and 
 draw the line A h. 
 
 3. At the parallel diftance of 14 min. H draw c d in¬ 
 finitely, and make B c and C d, each equal to 17 min. 
 and draw the line c d. 
 
 4. At the parallel diftance of 4 min. ~ draw D g /'rig. xviii. 
 infinitely, and make C e and D f each equal to 30 min. 
 
 and draw the line e f 
 
 5 
 
 y. At 
 

 
 Of the Geometrical Conjiruclion of 
 
 T. At the parallel diftance of 45- min. draw the line 
 E i inhnitely, and make D g and E /j, each equal to z6 
 min. and draw the line g h. 
 
 6. At the parallel diftance of s min. draw F /e / infinite¬ 
 ly, and make E i and F k, each equal to xy min. and 
 draw the line i k, alfo make F I equal to 30 min. 
 
 7. At tlie parallel diftance of y min. draw G n infinite¬ 
 ly, as alfo H 0, and make G n, and H 0, each equal to 
 3 7 min. 
 
 8. Draw the line / ;z, and divide it into 4 equal parts, 
 and defcribe the triangle ?i m /, making the lides n m and 
 m /, each equal to three parts of In, and the point m is 
 the center of the arch n, i, /. 
 
 9. At the parallel diftance of 6 min. draw IP^r infi¬ 
 nitely, and make Iq equal to 39 min. and and make 
 I r equal to (14 min. 
 
 10. Draw the line q 0, and with the diftance 0 q on 0, 
 defcribe the arch q P, and on q the arch 0 P, interfedling 
 each other in the point P, which is the center of the 
 arch q, 7, 0. 
 
 11. At the parallel diftance of 8 min. draw K S infi¬ 
 nitely, and make K S equal to 1 r, and draw r S ; alio 
 make S t equal to i rain. 
 
 12. At the parallel diftance of 3 min. draw L .r / 
 infinitely, as alfo M P, at the parallel diftance of j min. 
 and make L s and M P, each equal to (18 min, and draw 
 the line « P. 
 
 13. Draw the line t y, and divide it into two equal 
 parts in w, and on /, with the diftance / vj, delcribe the 
 arch nv u, and with the liimc diftance on la defcribe 
 the arch t ti, interfecling the former in which is the cen¬ 
 ter of the arch t w, and in the fame manner on x, clef- 
 
 Fig. xviii. j-j-jPg the arch w y. 
 
 14. At the parallel diftance of 6 min. draw N S in¬ 
 finitely, as alfo OT, at the parallel diftance of 2 min. 
 and make N S and O T, each equal to yd min. and 
 draw the line S T. 
 
 I y. Draw the line P S, and divide it into two equal 
 parts in V, and with the diftance P V on P defcribe the 
 arch V Q_, and with the fame diftance on V defcribe the 
 arch P interfe(fting the former in Q., whereon, with 
 the fiune diftance, defcribe the arch P V, and in the lame 
 manner on R, the arch V S alfo, which will complete 
 the profile, or geometrical elevation of the architrave, 
 freize and cornice, as required. 
 
 P R O n L E NJ 
 
Vlans and Uprights. 
 
 6i 
 
 Problem X. 
 
 To delineate the triglyphes of the dorkh order. 
 
 This ornament is feldom ufed in any order belides the 
 dorick, and is always placed in the freize exaftly over the 
 column. The height of this ornament is always equal to 
 the height of the freize wherein ’tis placed, (the capital 
 excepted) and the breadth to hall the diameter of the 
 column at the bale. In every triglyphe are y parts, viz 
 two entire glyphes or channels (as s m) meeting 'in an 
 angle, two femi-glyphes, as « and three interftices or 
 fpaces, as ^ /, &c. To delineate this ornament you 
 mult, ^ 
 
 I. Take ly min. and place from D to Z, and from E 
 to h, and draw h Z. 
 
 z. Divide D Z and E h, each into 6 equal parts and 
 draw the lines ah, ah, &c. 
 
 g. Set z min. from htox, and from E to and draw 
 the line x. 
 
 4. On AT, with the diftance x 0, defcribe the quadrant 
 0 n, and with the fame opening on m the femicircle 0 n 0 
 
 Hence it appears, that the triglyphe muft be divided in¬ 
 to iz equal parts, of which two mult be giVen to each 
 entire channel, as well as to the fpaces between, and one 
 to eacli femi-channel, at the extreams. 
 
 y. Continue the lines h T, a h, &c. through the lift of 
 the architrave towards 000, &c. and draw the line a a 
 parallel to C e, and p p. ^ ^ 
 
 6 . Make the parallel diftance of jO p, equal to i min - 
 and ^ ^ to 4 min. ' ^ 
 
 Lajily, if right lines be drawn from the points of inter- 
 feclion r r, &c. towards the points c c c, &c. (which 
 are in the midft of the lift) till they meet the line p p 
 they will truly form the guttie, or drops, and complete 
 the whole, as required. 
 
 Thefe guttae, oi drops, are made either in fliape of the 
 fi uftumof a cone, or pyramis, and oftentimes exact cones 
 pr pyraments. 
 
 When triglyphs are placed throughout an entablature, 
 the empty fpaces between muft be exaftly fquare (and are 
 call^ metops). From whence it happens, that in many 
 ftiuctuies the triglyphes are left out, on account they can- 
 
 ^ R not 
 
62 
 
 Of the Geometrical Conjlruflion of 
 
 not be fo diftributed, as to make tlie empty fpaces, or 
 metops, exabtly fquare. Thefe metops are oftentimes en¬ 
 riched with oxes fcullSj fruit, flowers, &c. according to 
 the nature of the building wherein they are intro¬ 
 duced. 
 
 Problem XI. 
 
 To defer the the upright atid inverted cima, or epnaife, 
 vulgarly called ogee. 
 
 I . Of the upright cima. Fig. A R. 
 
 Plate Vll. 
 
 Pr aft ice. 
 
 I. Draw the right line a m, and bilTe< 3 : it in n. 
 
 a. On m, with the diftance m n, deferibe the arch nr, 
 and alfo on n the arches m r and n o. 
 
 3. With the lame opening on a, deferibe the arch no. 
 
 Lajlly, The points 0 and r, are centers whereon you 
 may deferibe the arches m t n and n i a, which will com¬ 
 plete the upright cima, as required. 
 
 II. Of the inverted cima. Fig. A D. 
 
 Plate VII. 
 
 Praftice. 
 
 I . Divide the projefture given to the cima, as a h, in¬ 
 to 6 equal parts. 
 
 a. Make m I and g i, each equal to ~ of a I, and draw 
 the line i m. 
 
 3. Bilfeft imxn'k, and then deferibe the cima, as in 
 the preceding, and the inverted cima will be completed, 
 as required. 
 
 Problem XII. 
 
 To delineate the geometrical upright of any pillafer, 
 or column, 'with its entailature. 
 
 I. Of 
 
Tlans and Uprights. 
 
 1 . Of a pillaftcr. 
 
 ^3 
 
 Plate VII. 
 
 Practice. 
 
 I. Draw the line I H for the centeral line. 
 
 а. By the firll of problem IX. hereof^ delineate the 
 bafe G. 
 
 3. Make K L equal to the affigned height of the pillaf- 
 ter, via. 7 diameters, &c. and through the point L draw 
 B C parallel to E F, and make K F, K E, L C, L B, 
 each equal to the femidiameter of the pillafter, via- 
 30 min. 
 
 4. Draw the right lines B E and C F, and then will 
 the body of the pillafter be completed. 
 
 S- By the fecond of problem IX. hereof, delineate the 
 capital A; and by the third, the architrave, freize and cor- 
 nilh M D N, and then will the whole be completed, as 
 required. 
 
 II. Of a column. 
 
 Plate VII. 
 
 Pradlice. 
 
 I. Draw the centeral line A Q,. 
 
 By the fir ft of problem IX, hereof, delineate the 
 bafe B. 
 
 3. Make C I equal to the affigned height of the fliaft 
 of the column, viz. 7 diameters, &c, and through the 
 point I draw the right line L I K at right angles to the 
 centeral line A 
 
 4. Divide C I into 3 equal parts, and fet up one from 
 C to F, and through the point F draw the right line 
 GFH. 
 
 y. Make CD, C E, F H and F G, each equal to the 
 femidiameter of the column at the bafe, viz. 30 min. 
 and draw the right lines D H, and E G, parallel to C F. 
 
 б . Make I K and I L, each equal to the femidiame¬ 
 ter at the capital or head of the ffiaft, via. ad min. 
 &c. 
 
 7 On 
 
Of the Geometrical Confirnhlion of 
 
 7. On F defcribe the femicirclc G « ^ H, and make the 
 chord line a a, equal to L K and parallel to G and 
 draw the riglit lines L a and K a. 
 
 8. Divide tlie arches a H and a G, into ant" number 
 ol:' equal parts (the more the better) fuppole 4^ as in tire 
 diagram at the points n m 0 G, See. and draw the lines 
 n n, m yn, and 0, 0. 
 
 9. Divide F I into the fame number of equal parts, as 
 a G or a H, which in this example is 4, at the points 
 1,1, 3, I, and through the points i, 2 and 3, draw the 
 right lines W i X, T i V and R 3 S, at right angles to 
 the centeral line A Q,. 
 
 10. Upon the points « and n, ercdl the perpendiculars 
 71 R, n S, or at the dillance of i 71, draw the lines n R and 
 n S parallel to F 1 , and they will interfecl the line R 3 S 
 in the points R and S. 
 
 11. At the dillance of i vi, draw the parallels mH 
 and 771 V, and they will interlefl; th.c line T 2 V in the 
 points T V. 
 
 II. At the diftance of 3 0, draw the parallels o W, and 
 0 X, and they will interfecl the line W i X in the points 
 W X. 
 
 Laftly, lines being drawn from G to L, and from FI to 
 K, though the feveral points of interfecFion W T R, and 
 S V X, fliall truly form the diminlhing (or upper) part 
 of the lliaft, as required. 
 
 To which being added the capital and entablature, as 
 before taught, the whole will be completed, as requi¬ 
 red. 
 
 iV B. That in confideration, as the upper part of the 
 fliaft of every column is fo much leller than the up¬ 
 per part of a pillafter, by fo much as the diminu¬ 
 tion of the column is, as generally in the tufean (•, the 
 dorick 7, the ionickj, the corinthian and the com- 
 polita j, of their diameters at the bafe, therefore when 
 you are to delineate any pillafter, &c. with its entabla¬ 
 ture, from any ofthe geometrical elevations, at the end 
 hereof, you muft add to the projedlure of every 
 member, half the diminution of the column, and there¬ 
 by every member will have its true projedlure. 
 
 Problem 
 
Vlans and Uprights. 
 
 <^5 
 
 Problem. XIII 
 
 To delineate the geometrical upright of any wreath'd 
 wa'ced or twifed columns. ’ 
 
 Thefe kind of columns may be defcribed divers wavs 
 but none better than the following. ’ 
 
 Plate VII. 
 
 Practice. 
 
 o r problem, delineate the corinthian rig. iir. 
 
 lhaft B O D N, and make B A equal to B D. 
 
 2. Diaw the right line A D, and on the point A with 
 any ladius, defcribe an arch as C Z, which divide into it- 
 equal parts at the points i, 2, 3, 4, &c. 
 
 3. Lay a ruler fi'om A to the feveral points i, 2 3 4, <• 
 
 &c. and draw right lines to the fide of the column D b’ 
 as to the points n, n, n, &c. 
 
 4. From the feveral points n,n,n, 8 cc. draw the right 
 fines nm, nm, nm, &c. parallel to the bafe D N 
 
 J. On N with the diftance N m, defcribe the arch 
 and with the fiime opening on m, defcribe the arch N i 
 interfering the former in the point i, which is the center 
 of the arch N m. 
 
 6 . Perform the fame operation at the feveral divifions, 
 and thereby you will complete the fliaft as required. 
 
 Plate VII. 
 
 Shewing how to perform the aforefaid operation a dif¬ 
 ferent nuaj from the foregoing. Fig.iv. 
 
 Praftice. 
 
 m r P^'^‘^^^ding problem, delineate the ionick 
 
 lhatt L R 1 and make P E equal to one third of F P, 
 and draw the right line F E. 
 
 2. Wkh the diftance E F, on E defcribe the arch F V, 
 and on F the arch E V, and alfo on V the arch E i, 2, 
 L 4 ^ L d, 7, 8, 9, 10, II, F. 
 
 S Di- 
 
 3. Di- 
 
66 
 
 Of the Geometrical Conjtruciion of 
 
 3. Divide the arch E i, r, 3, &c. F, into iz equal 
 prats, at the points i, z, 3, 4, f, 6 , 7, 8, 9, 10, ii, 
 and from them draw right lines parallel to the bale P 
 ’till they interfedl the column in the points n, n, n, &c. 
 and 0, 0, 0, &c. 
 
 4, Divide each of the divifions n, n, &c. and 0, 0, 0, 
 
 &c. as or P n, into 4 equal parts, and with the 
 
 dillance of 3 of thole parts, deferibe the feveral arches 
 therein on the points P, n, n, n, &c. and Q_, p, 0, 0, 
 &c. interfedling each other in r, r, r, &c. which points 
 of interfedlion are the centers of the feveral arches that 
 compofe the column, and being deferibed will com¬ 
 plete the lhaft, as required. 
 
 Plate VII. 
 
 Shew the like operation in Jmall columns. 
 
 Pradlice. 
 
 I. By the preceeding problem delineate the dorick 
 lhaft S T G I, and make I K and G H, each equal to G I, 
 and draw H K, and the diagonals K G and H I, inter¬ 
 fering each other in n, whereon deferibe the arch K L 
 
 z. Make the triangle G m H, equal to the triangle 
 I « K, and on m deferibe the arch H G. 
 
 3. Make K M and H L equal to H K, and draw the 
 line L M and the diagonals M H and L K, interfering 
 each other in u, the center of the arch H L. 
 
 4. Make the triangle K w M equal to K « M, and on 
 w deferibe the arch K M, and fo on with all the others, 
 and thereby the whole will be completed as required. 
 
 The lhafts fig. VI, VII, and VIII. are the feme Ihafts 
 completed, whereby their effer may be adjudged. 
 
 Problem XIV. 
 
 Plate VIII. 
 
 How to divide the Ireadth of any pillafler into its 
 flutes and fI Lets, and to delineate the geometrical upright 
 of the fame- 
 
 I. The general received proportion for dividing the 
 breadth of a pillafter, is to divide every pillafter into 
 feven flutes and eight fillets, and that the breadth of e- 
 4 very 
 

Tlans and Uprights. 6-j 
 
 very fillet contain ^ part of the breadth of a flute, and 
 thereby the breadth of every pillafter fo fluted, is divid¬ 
 ed into a9 equal parts, viz. eight equal parts contained 
 in the eight fillets, and twenty one equal parts in the feven 
 flutes, each containing three, and thereby every flllet is pig. xix, 
 equal to ^ of a flute as aforefaid. This being well under- 
 ftood, we ll now proceed to the geometrical conftrubtion 
 thereof. 
 
 Plate VIII. 
 
 Let the line V G he the given breadth of a pillafter, 
 to he divided into its flutes and fillets, as aforefaid. 
 
 I. Draw a line at pleafure, as A B. 
 
 1. With any fmall opening of the compafles, fet off 19 
 times of that opening, beginning at any part thereof, as 
 at B, and ending at A, as in the figure. 
 
 3. Having thus fet off ^9 equal parts on the line A B, 
 the next work is to make an equilateral triangle there 
 from, which thus perform. 
 
 On the point A, with the diltance A B, defcribe the 
 arch B a, and with the fame diftance on B, defcribe the 
 arch A a, interfering the former in the point C, and 
 draw the lines A C and C D. 
 
 4. From the angle, or point C, draw right lines through 
 all the X9 divifions marked i, x, 3, &c. and continue 
 them infinitely, and thus have you prepared in efter an 
 inftrument that will at once divide the breadth of any pil- 
 lafler that may be given, as fhall appear by the example 
 in hand. 
 
 y. Take the given line F G in your compafles, and fet 
 that diftance from C to E, and from C to D, and draw the 
 line D E; and becaufe the figure is equilateral, therefore 
 D E is equal to the given line F G, and by the lines C 0, 
 
 Co, &c. drawn through the x9 divifions, is divided in¬ 
 to x9 equal parts alfo, which is the divifion of the pil¬ 
 lafter required. 
 
 Operation. 
 
 j. Eredl the perpendicular E H, which reprefents one 
 fide of the pillafter. 
 
 X. The firft fillet being equal to 5 of D E, therefore 
 at the diftance E h, draw h h parallel to EH, and it fhall 
 be the firft or outfide fillet. 
 
 3. As 
 
68 
 
 Fig. XIX. 
 
 Of the Geometrical ConfiruEtion of 
 
 3. As every flute is equal to three fillets, therefore 
 number three equal parts from h to c, and draw c c 
 parallel to b b, and it fliall be the firlt or outlide flute. 
 
 4.. At the diltance of one divifion from c to d, dvnwdd 
 for the next fillet, and alfo at the diftance of three divili- 
 qns tfom d to e draw e e for the next flute, and in the 
 lame .manner, taking one divifion for a fillet and three 
 for a flute, you will complete the flutes and fillets of the 
 pillaller, as required. 
 
 K B. The depth of every flute in the pillafter is ^ the 
 breadtli of the flute, therefore to defcribe the cir¬ 
 cular termination, fet up ; of the breadth of the 
 flute from x to s and that will be the center of the 
 curve that terminates the flute c c, and the like of all 
 others. 
 
 'Tis to be obferved (as I faid before) that in refpedt 
 to the figure being equilateral, the breadth of any 
 pillaller may thereby moll readily be divided, be the 
 fame but the tenth of an inch, or 1000 feet, &c. and 
 therefore of univerliil ufe. 
 
 rig.XIX. "pjic line M N is tlie breadth of a fmaller pillafter, 
 wliich is divided in the fame proportion and by the ve¬ 
 ry lame rule, and is inferred to iliew the reafon of the 
 figure without any more words, to which I refer you. 
 
 Problem XV. 
 
 T0 dtvide the bajis, or plan of the fjaft of a column 
 into its Z4 flutes, arid aq. fillets. 
 
 The number of flutes were formerly limited to every 
 order, the dorick being allowed xo, and the ionick X4. 
 But that limitation has been difpenfed with, with di- 
 L'crs of our modern architefls. In this example ’twill 
 be fufficient to divide a femicircle, or the femi-bafis of 
 the lliaft, inllead of the whole, the laft being but the 
 firft repeated. 
 
 Pratftice. 
 
Plans and Uprights. 
 
 Pradlice. 
 
 I. Let B D be the diameter of the balis of a column 
 given. Jig, XX. 
 
 %. Divide the fame into two equal parts in A, and 
 thereon, with the dillance A B, deferibe the femicircle BCD. 
 
 3. Divide the fame into two quadrants by the perpen¬ 
 dicular A C, and divide each quadrant into ir equal parts, 
 and draw the lines a, a, a, a, &c. through the fame. 
 
 And thus is the femi-balis prepared for the defeription 
 of the flutes and fillets. 
 
 4. Divide any of the i x parts (as B ^ ) into eight e- 
 qual parts. 
 
 y. Take three of thofe eight equal parts in your com- 
 pafles, and on thofe points where the lines a, a, a, &c. 
 interfefl the femicircle BCD, deferibe the feveral ar¬ 
 ches z, z, z, z, &c. which lhall be the flutes, and in¬ 
 tervals of the fillets, as required. 
 
 Problem XVI. 
 
 To divide the hafts, or plan of the fjaft of a colimin, 
 into its 14 flutes without fillets, as is ufual in the do- 
 rick order. 
 
 I fliall here (as in the laft) make ufe of the femi-bafis ^g. xx. 
 only. 
 
 Pradlice. 
 
 I. Complete the femicircle BCD, and divide the fame 
 alfo into i x equal parts, by tJie lines n, n, n. See. 
 
 X. Divide any one of thofe parts, into eight equal parts 
 as the part i and x. 
 
 3. From the feveral points where the lines n, n, n. See. 
 interledl the lemichxle, on thole feveral lines fet off two rig.xx, 
 of thole eight parts, as at the points 0, 0, 0, 0, See. which 
 are the centers of each flute. Therefore on thole points, 
 with the dillance 0 i or 0 x, &c. deforibe the feveral 
 arches, and they will complete the flutes of the femi- 
 bafis, as required. 
 
 T 
 
 Problem 
 
70 
 
 Of the Geometrical Conjlruclion of 
 
 Problem XVII. 
 
 T0 defcrihe on a paper drawing, wall, &c. the geo¬ 
 metrical upright of a column, with its flutes and fil¬ 
 lets. 
 
 To defcribe the geometrical upright of a column, is 
 to Ihew ill what manner the flutes and fillets diminilh 
 in their breadth, as they approach the extream parts of 
 the column. 
 
 Pradticc. 
 
 If from the interfedlion of the flutes and fillets, you 
 draw the lines r, r, r, r, &c. perpendicular to B D, 
 and complete their terminations with circular lines, as 
 in the figure, they will complete the geometrical upright 
 of that part, as required, and every flute and fillet have 
 its due breadth, according to the rules of perfpedlive. 
 
 Problem XVIII. 
 
 To defer ihe [in the afore [aid manner') the geometrical 
 upright of a column, with its flutes only, as often ufed i7i 
 the dorick order. 
 
 Pradlice. 
 
 I. It from the interfedlion of tlie flutes in the femi- 
 circle B E, D, you draw the lines i, s, s, &c. perpendicu¬ 
 lar to B A D, and complete their terminations with circu¬ 
 lar lines, as in the figure, they will complete that geome¬ 
 trical upright, as required. 
 
 Problem XIX. 
 
 To divide the hafe, or plan of the fiaft of a column 
 into its ao or zy futes, according to Vitruvius. 
 
 Pradlice. 
 
 I. On A defcribe the bafe of the Iliaft, as the circle 
 B < 3 ! Z' D E. 
 
 a. Divide the circumference into lo or equal parts 
 by the lines n, r, r, r, &c,. 
 
 3. Make 
 
 
Plani and Uptights. 
 
 3. Make i a and i h, each equal to 7 t m, and draw the 
 right line a h. 
 
 4. Complete the geometrical Iquare a d c h, and draw 
 the diagonals a c and d h, interletting each other in n, 
 the center of the flute a 0 h. 
 
 S- On A, with the diftance A n, deferibe the circle 
 n r r r, &c. interfetling the lines r, r, r. See. which are 
 the X4 centers of the 24 flutes, whereon, with the radius 
 n b or a n, you may complete the whole, as required. 
 
 Problem XX. 
 
 T0 divide the hafe, or plan of the fjaft of a column 
 into its zo or Z4 flutes, according to Vignola. 
 
 rratlice. 
 
 I. On E deferibe the bafe of the fliaft, as the circle 
 Kdb^Y). 
 
 z. Divide the circumference into zo, or Z4 equal parts 
 by the lines e, e, e, &c. 
 
 3. Make i h and i d, each equal to ^ i h, and draw the 
 right line d b. 
 
 4. Complete the equilateral triangle dab, and then will 
 the angle a be the center of the flute dnb. 
 
 S- On E, with the diftance E a, deferibe the circle 
 e e e, &c. interfering the lines e, e, e, &c. in the points 
 e, e, e, &c. which are the Z4 centers of the Z4 flutes, 
 whereon, with the radius a b oc a d, you may complete 
 the whole, as required. 
 
 N. B. That although both the fliafts in thefc examples 
 are divided into Z4 flutes, yet you are to underftand 
 that neither Vitruvius or Vignola made ufe of any 
 more than twenty; therefore if you are willing to fol¬ 
 low their rules therein exablly, you mull divide the 
 circumference of the fliaft into zo parts, inftead of 
 Z4, and then proceed in all other refpefts, as in the 
 preceding problems, and thereby you will complete 
 the whole, as required. 
 
 Problem XXL 
 
 To divide the bafe of the Jlsaft of a column, into its 
 cabled flutings. 
 
 4 
 
 Pradlice. 
 
72 
 
 Of the Geometrical Conflrudion of 
 
 Praftice. 
 
 Fig. XXIII. I. On A defcribe the bafe of the ftaft, as the circle 
 'a a a, &c. 
 
 ^ 2. By problem XV. hereof, delineate the flutes and 
 
 fillets thereof 
 
 3. On 0 0 0, &c. with the radius 0, a, 0, a, &c. defcribe 
 the arch r a $, &c. and thereby you will defcribe the 
 cabled fluting, as required. 
 
 I Problem XXII. 
 
 To dpvide the hafe of the foaft of a column into its 14 
 flutes and 24 fillets, after the manner of the columns 
 within the Pantheon. 
 
 Pradlice, 
 
 Fig.XXIV. I. Defcribe a circle reprefenting the bafe of the fliaft, 
 and divide the circumference thereof into 24 equal parts 
 by the lines a, a, a-, &c. 
 
 2. Divide each part into y equal parts, of which give 
 4 to every flute, and one to each fillet. 
 
 ■ 3. Make the depth of each flute, equal to the breadth 
 of every fillet, and then will the whole be completed, as 
 required. 
 
 Fig. XXV. is the plan of the dorick lliaft cut into cants 
 inftead of flutings withou any cavity, firft taught and 
 pracficed by Vitrumus, which I here infert, to fhew the 
 young fludent what a great variety there is contained in 
 the form and manner of fluting columns. 
 
 Problem XXIII. 
 
 To defer He the ionick valuta according to the antique 
 7 na 7 iner. 
 
 P L A T E IX. 
 
 Pradlice. 
 
 Fig- 1 - I. Delineate the Abacus B A, and from the point x 8 i, 
 let fall the cathetus, perpendicular, and delineate the hol¬ 
 low of the voluta B B, the ovoloBD, the aftragal B E, and 
 the cinbture or annulet B F. 
 
 2. Draw 
 
/'lit!,' J-. WII 
 
Tile ±^ye y of y ’’.'oliLta- 
 
73 
 
 T^luns dhd Uprights] 
 
 1. Draw the right line F B lb as to pals through the 
 middle of the aftragal B E, interfeaing the cathetus in 
 the point i 5. 
 
 3. On the point i 3, with the dillance i 3 Y, defcribe the 
 circle or eye of the volute Y X Z, and draw the geometrical 
 fquare YXTZ. 
 
 4. Billedl: 1 X, XT TZ and Z Y in the points 1, 3^ 
 4, and draw the geometrical fquare 1234, and alfo the 
 diagonals i 3 and z 4. 
 
 5. Divide each diagonal into 6 equal parts at the points 
 s, 9, II, n, 6 , 10, IZ, 8. 
 
 6 . Make i a equal to ^ of i 1, as alfo z c, 3^4 h, 
 ^ i, 6 n, ^ V, k, <) 0, 10 m, t, 12 and thus will 
 you have divided the eye of the volute into its proper 
 centers^ on which you may defcribe it as follows, viz. 
 on the point i, with the opening i,z8i, defcribe the 
 arch z8j- B, and on the point a the arch n m, alfo on the 
 point z the arch B C, and on c the arch m w ; likewife 
 on the point 3 the arch C F, and on d the arch w E, and 
 fo by removing to the other centers 4 h, &c. you will 
 complete the whole voluta, in the moll elegant manner 
 as can be delired. And that the young Undent may have 
 a perfedl idea of the centers thereof (which in number 
 are zy) I have in fig. II. defcribed the eye of the volute 
 at large, wherein the numerical figures denote the cen¬ 
 ters of the exteriour line, and the fmall italick letters the 
 centers of the intcriour, which in an inllant will enable 
 him to delineate the fame with great eafe and delight. 
 
 Sect. IL 
 
 Of the Derivation, Proportion, Diminution and In¬ 
 ter columnation of the Tufcan, Dorick, lonick, 
 Corinthian and Compojite Orders of Archite^ure. 
 
 1 . Of the Tufcan order. 
 
 ^ I 'HE Tufcan or rullick order (faith Vitruvius) is the 
 moll fimple and llrongell of all the orders of ar- 
 chitedlure, it hath no ornaments and but few mouldings. 
 This order was firll made by the Afiatic Lydians, who 
 
 V are 
 

 Of the Derimtion] Troprtion, Szc. 
 
 (lie laid to be the firfl: that inhabited Italy, and brought it 
 into that part, called Tufcana or Tufcany, and from thence 
 was called the Tufcan order. 
 
 And altho this order is of all others the moft 2>lain 
 and hmple, yet many noble ftruaures have been built 
 therewith, as the ports and entrances into cities, amiihi- 
 theatres, bridges, &c. and particularly that famous co¬ 
 lumn of the Trojans, that of Antoninus at Rome, and 
 likewife that of Theodofms at Conjlantinople, which are 
 all remaining to this dtay. 
 
 The proportion, diminution and intercolumnation of 
 this rural order is as follows. 
 
 I. The lliaft only without its bafe and capital is in 
 length lix diameters of the fhaft’s bafe, and the height of 
 the bafe and capital each a femidiameter thereof 
 
 a. The entablature ol this order is feldom lefs than - of 
 the lhaft’s height. 
 
 3. The pedellal hath two diameters of the lhaft’s bafe 
 lor Its height, and the lhaft at the upper part diminilh- 
 es ~ of its diameter at the bafe. 
 
 4. The intercolumnation of this order may be made 
 very large, by lealon the architrav^e is generally made 
 ol wood, but the moft ufual is about 4 diameters of 
 the lhaft at the bafe. 
 
 II. Of the Dorick order. 
 
 The Doiick oider is of all others the moft grave and 
 mafculine, and the moft agreeable to nature. Scamozzi 
 calls It Herculean afpedt, in regard to its excellent propor¬ 
 tion. This order had its original and name from thoTori- 
 ans, a Grecian people of ylfia, or as Ibme fiy, from ‘Do¬ 
 rns King of Achafis, who is fiid to be the lirlt that built 
 at Argos, and dedicated a temple of this order to Juno. 
 
 The proportion, diminution and Intercolumnation of 
 tuts noble order, is as follows. 
 
 I. The lhaft, exclulive of the bafe and capital (when 
 alone) is in length fev^en diameters, but when in porticos 
 and mural work but fix. 
 
 of the capital is a femidiameter of the 
 lhalts bafe, as alfo the Attick bafe, &c. when ufcd here- 
 
 I in. 
 
75 
 
 of the Tufcan, Doricf See. Orders of ArchiteElure. 
 
 in. For you muft note, that this order was anciently made 
 without any bafe, as may be feen by the geometrical 
 profiles ot the Theatre of Marcellus, the Bath of ‘Dio¬ 
 cletian, &c. at Rome, at the end hereof 
 
 3. The entablature is generally two diameters in hei<^ht, 
 and IS oftentimes enriched in the freize with triglyphes 
 and metops. The lhaft of this order is oftentimes fluted 
 with a fliort edge without any fillets, as laid down by 
 Talladto and Vignola, in thirpreceding problems of feft. I. 
 hereof And as I faid before, that the ancients, never 
 ufed any bale to this order, fb tis alfb to be underffood 
 of pedeflals ; therefore when any are ufed herein, Talla- 
 dio allows their height to be two diameters, and t of the 
 lhaft’s bafe. ' 
 
 4. The diminution of the fliaft is i- of the fliaft’s dia¬ 
 meter at the bafe. And the intercolumnation of this order 
 is three diameters, except at fuch times when the diftri- 
 bution of the triglyphes and metops require fomething 
 more or left. ° 
 
 III. Of the lonick order. 
 
 The lonick order is an exadl: mean proportion, between 
 the delicate and the robuft. Vitruvius compares it to a 
 matron decently drefs d. It was firft invented, or intro¬ 
 duced by Ion, in Ionia, a province in Afia, and ’tis faid 
 that the Temple of Diana, at Ephejus, was built of 
 this order. 
 
 The proportion, diminution, and intercolumnation of 
 this dece 7 it feminine order, is as follows. 
 
 I. The fliaft with its bafe and capital, were anciently 
 but 8 diameters, which by the moderns was thought too 
 little, and therefore to give it proper ftature, they added 
 one diameter, lb that it now contains 9 diameters in height. 
 The fhaft is fluted with xq. flutes, with fillets between, 
 whole breadths are equal to | of a flute. 
 
 X. The entablature is \ of the altitude of the column, 
 and its cornifli is always adorn’d with denticules. 
 
 3. The height of the pedeftal is two diameters and ~, 
 and its intercolumnation two diameters and which is 
 the mofl; elegant manner of intercolumnation, and by Vi¬ 
 truvius is called Eujiillos. 
 
 Laftly, The diminution of the lhaft is f of the diame¬ 
 ter at the bafe of the fhaft. 
 
 IV. Of 
 
OJ the Derivation, Proportion, &c. 
 
 IV. Of the Corinthian order. 
 
 The Corinthian order is the very pride and delicacy of all 
 the other orders. It was firft defign'd by an architeft of A- 
 thens, and executed at Corinth, a noble city of Telopon- 
 nefe, or Morea, from whence it had its original and name 
 of Corinth in order. 
 
 The proportion, diminution, and intercolumnation of 
 this heatifid order, k as follows. 
 
 I. The llraft with its bafe and capital is 9 diameters and 
 a haUi, and fometimes 9 and and oftentimes 10 dia¬ 
 meters in length. If the draft be fluted, the flutes muft 
 be made according to problem XV. fedl. I. hereof 
 
 r. The height of the capital is one diameter of the 
 draft at the bafe, of which the abacus irrufl; be a fixth, 
 or feventh part, and the reirraining quantity being divided 
 into 3 equal parts, the two lowerirrofl; is the true height 
 of the fil'd: and fecond toure of leaves, and the third or 
 uppermoft part being divided into two equal parts, the 
 upper part of thofe two parts drall be the extreams of the 
 volutas and the lower the cauliculi. 
 
 5. The height of the entablature is ~ of the column, 
 including the bafe and capital, except when applied to 
 great and magnificent buildings, as the Roman Tantheon, 
 &c. 
 
 4. The height of the pedeftal mud: be of the alti¬ 
 tude of the coluirrn, and the diminution of the draft \ of 
 the diameter at its bafe. 
 
 5". The intercolumnation is two diairreters and j, as 
 in the preceding order of the lonick. 
 
 V. Of the Compofite order. 
 
 The Compofite order, is of Roirran extraftion, and by 
 irrany called the Italian order, and oftentimes the Roman 
 order. ’Tis compofed of the lonick and Corinthian orders, 
 and therefore is called the coirrpofed order. 
 
 The proportion, diminution and intercolumnation of 
 this order, is as follows. 
 
 I. The draft, with its bafe and capital, is ten diairreters 
 in length, or height; and its entablature ', or 7, thereof 
 
 Its 
 
Oj the Tufcun^ Dorick, See. Orders of ^rchiteHure. yy 
 
 its diminution at the head of the ftaft is f of the dia¬ 
 meter at the bafcj and its intercolumnation one diame- 
 tei' and or i. 
 
 The height of the pedcftal is generally equal to - of 
 the column’s altitude, and its bafe is either attick, or a com¬ 
 pound ol the attick and ionick. 
 
 And altho thefe pi opoi tions of all the five orders are 
 thus eftablillied, yet not with fo great a ftriftnefs, but 
 that the architea may vary therefrom, upon juft oc- 
 cafions, as tlie grandeur and conveniency of a building 
 may require. ° 
 
 
 Sect. III. 
 
 Of ^rchite&onical Axioms and Analogies. 
 
 I. Of doors. 
 
 'pHat the height of all doors be double their breadth. 
 
 That doors in general be proportional to the mag¬ 
 nitude of the rooms. 
 
 That the breadth of inner doors be never left than 2. 
 feet i, nor more than 6 feet. 
 
 That the doors of the ad ftory be placed exaftly over 
 the doors of the firft, and the like of the 3d, &c. 
 
 That an aich of brick or ftone be turned over every 
 door,todifcharge tlie weight that preffes upon them, which 
 oftentimes ruines the ftruaure. 
 
 II. Of windows. 
 
 That the magnitude and number of windows be propor¬ 
 tional to the rooms that they are to illuminate. 
 
 That the height of every window in the firft ftory be 
 double its breadth, with the addition of t or 2- part 
 as found to be neceflkry. ^ ^ ^ 
 
 That the height of the windows in the ad ftory be ^ of 
 
 the fiift, and the height of the attick or 2d ftory ^ of 
 the fecond ftory. . or 
 
 That windows be not placed too near the angles of any 
 building, that thereby the ftrudlure be not weaken’d. 
 
 3 X 
 
 That 
 
Of ArchiteUonical Axioms and Analogies. 
 
 That over every window be turn’d an arch to difcharge 
 the weight that lies over them. 
 
 That no girder be laid over any door or window, but 
 always on the moft fubllantial part of tlic brick or Hone 
 peers, &c. that Iblid may reft upon folid. 
 
 That Venetian windows have their proportions, as fol¬ 
 low, VIZ. (fig. IX. plate ATI.) that the hcigiit A B be e- 
 qual to twice E F, and that G H and C D be each e- 
 qual to E. F. 
 
 That the centers of all pediments be placed down the 
 centeral line at the diftance of - the length ol' the co¬ 
 rona. So the point c of fig. X. plate VII. is the center 
 of that pediment, being the diftance of a, b, fet down 
 to c and the like of all others in general. 
 
 in. Of gates. 
 
 That the breadth of principal gates of entrance be ne¬ 
 ver Id's than 7 I'eet nor more than ix feet. 
 
 That the height of principal gates of entrance be never 
 Icfs than their breadth and )■, nor more than twice, which 
 is the bell proportion. 
 
 IV. Of halls. 
 
 That the length of halls, be not Ids than twice their 
 breadth, nor more than three times. 
 
 That the heiglit of halls, whofe cielings are flat, be not 
 lefs than of the breadtlr, or more than f of the length. 
 
 That the height of halls whofe cielings are arched be not 
 lefs than f-, nor more than ;■) of their breadth. 
 
 V. Of galleries. 
 
 That their fite be towards the North, on account that 
 the North light is the beft for painting, pictures, &c. 
 
 That the breadth of galleries be not lefs than 16 feet, 
 nor more than 14. 
 
 That the length of galleries be not lefs than p times 
 their breadth, nor more than eight at moft. 
 
 That the height of galleries be equal to their breadth, if 
 with flat cielings, but if arched, the breadth and j or j-. 
 
 Of antichambers. 
 
 That the length of all antichambers be equal to the hy- 
 pothenufe of a right angled plain triangle, whofe legs are 
 each equal to the breadth of the antichamber. 
 
 That 
 
79 
 
 Of Architecionical Axioms and Analogies. 
 
 That the breadth of all anticharabers be proportional to 
 the whole ftrudlure. That the height of antichambers 
 be not lefs than \ of the breadth, or more than I of the 
 length, when the cieling is flat, and when arched, to be 
 not lefs than f-, nor more than p' of their breadth. 
 
 VI. Of chambers. 
 
 That all principal chambers of delight be placed to¬ 
 wards the belt profpedls of the country, and if polTible to 
 the Eaft. 
 
 That the length of chambers never exceed the breadth 
 and t of the breadth; therefore the length may be the 
 breadth exactly, or the breadth and j or t. 
 
 That the height of all chambers of the firfl: ftory, 
 whofe cielings are flat, be not lefs than ~ of the breadth, or 
 more than ^ of the length. 
 
 That the altitude of chambers in the fecond floor be 
 
 of the firfl: ftory. 
 
 That the altitude of the chambers in the third floor 
 be “• of the fecond. 
 
 VII. Of Floors. 
 
 That the floor of every ftory in a building be truly le¬ 
 vel throughout, fo as to pafs out of one room into ano¬ 
 ther, without going up or down ftairs, as is common in 
 many buildings. 
 
 That the height of the level of thefirft (or ground) floor, 
 be never lels than one foot, nor more than four feet. 
 
 VIII. Of chimneys. 
 
 I. Of hall chimneys. 
 
 That the proportion of hall chimneys be as follows, viz. 
 Their diftance between the jaums from d to 8 feet; their 
 height from 4. feet ^ to y feet ; their projedlion from x 
 feet r to 3 feet at moft ; the breadth of the jaums from 8 
 to X4 inches or more, as occafion may require, according 
 to the order that the chimney is adorned with. 
 
 X. Of chamber chimneys. 
 
 That the proportion of chamber chimneys be as follows, 
 viz. their breadth from y to 7 feet, their height 4. feet 
 
 and projedlure 1 feet and 
 
 3. Of 
 
 1 
 
So 
 
 Of ^rchiteclonical Axioms and Analogies. 
 
 3. Of chimneys in Undies, &c. 
 
 That the proportion of chimneys in fludies be as fol¬ 
 lows, ^Jtz. their breadth from 4 to y feet at molt • rbeir 
 height 4 feet and proieclure x feet 1. 
 
 That the funnels of chimneys of chambers, or Undies 
 be not narrower than 10 inches, or wider than ly, which 
 is a good fize for kitchin chimneys. 
 
 IX. Of the funnels of chimneys. 
 
 That the funnels of chimneys be carried a fufficient 
 height above the ridge, that reflex winds may not repulfe 
 the fmoke. ^ 
 
 That tlie funnels of chimneys be not wide, whereby the 
 w ind may dm-e down the linoke into the room, or too 
 narrow, where it cannot have a free paflage. ^ 
 
 That the funnels of chimneys be truly perpendicular, 
 othcnvile the Imoke cannot freely pafs, and thereby will 
 be ofleniive. 
 
 lhat no timber, joill, &c. be laid nearer to the iaums 
 than one foot. 
 
 Tliat no trimming joilts be laid nearer than 6 inches to 
 the back of any chimney. 
 
 That the funnels of all chimneys have not any timber 
 as girders, joill, &c. laid therein, otherwife the buildin^' 
 will be in danger of being reduced to allies. ^ 
 
 X. Of joills, rafters, and girders. 
 
 That the greateft dillance that joills, or rafters, are laid 
 from each other, do not exceed ix inches, and quarters 
 14 inches. ^ 
 
 That no joill bear at a greater length than ix feet or 
 hnglc rafters more than 10 feet. 
 
 That the length of joills laid in the wall be not lefs 
 than 9 inches, and no girder be lefs than ix inches. 
 
 XL Of flair cafes. 
 
 l hat flair cafes be fpacious, light and eafy in afeent. 
 
 That the breadth of Hair cafes be not lefs than 4. feet 
 or more tlian ix feet. 
 
 That the height of Heps be never lefs than 4 inches, or 
 more than 6 . 
 
 That the breadth of Heps be never more than 18 inches, 
 or lels than 11 inches. 
 
 xn. Of 
 
Of rchiteBonical Axioms and Analogies. 81 
 
 XII. Of materials^ &c. 
 
 I. That money and materials be always ready from the 
 beginning, or laying of the foundation, to the turning of 
 the key when the whole is completed. 
 
 а. That great care be taken in the goodnefs of founda¬ 
 tions, and that they be truly level. 
 
 3. That the thicknefs of all foundations be double to 
 the inhftent wall. 
 
 4. That the moll heavy materials be imploycd in the 
 foundations. 
 
 y. That all walls diminifli in thicknefs, according to 
 the nature and height of the ftruriure. 
 
 б . That every wall be perpendicular. 
 
 7. That fuch bricks as are not well burnt, be not ufcd 
 in any building. 
 
 8. That the depth of all fabricks in the ground that have 
 cellars, vaults, &c. be j of the whole height, and thofe 
 that have no cellars to be ~ of the height. 
 
 9. That the kitchin be fpacious and light, and as re¬ 
 mote from the parlor as poITible, and to be under ground ; 
 as alfo the pantry, bake-houfe, ftill-room, buttry, dairy, 
 and fervants offices in general. 
 
 10. That corniffies do not projeft too fir out from the 
 building, whereby the windows be darken’d. 
 
 II. That ol all kind of arches none is fo Ilrong as the 
 femicircle. 
 
 I a. That the depth of all rufticks be never more than 
 I foot, nor lefs than 9 inches. 
 
 13. That the thicknefs of pillalters, of doors and win¬ 
 dows, be not more than ^ of their aperture, nor lefs 
 than j. 
 
 14. That the projedlure of pillalters in general, be ~ of 
 their thicknefs. 
 
 I y. That the roofs of all buildings be not too heavy, or 
 too light, and that the interiour walls fupport part of the 
 fame. 
 
 16. That convenient citterns be well placed, plenti¬ 
 fully to furnitli every office with water, and that proper 
 machines be made to raife the fame therein. 
 
 hajily. That convenient drains, to carry away foil, &c. 
 be well contrived, and fecretly placed, with vents to dif- 
 charge the noifome vapours- 
 
 y 
 
 Sect. 
 
Sect. IV. 
 
 Plate X. 
 
 Of the Defcription and life of an infpeBional plain 
 Scale, for delineating x^rchiteciure, Gardening, Sic, 
 
 HIS fcalc being defigncd but for the drawing of ar- 
 chitethirc and gardening, it is therefore made to fuch 
 a breadth and length as is hiitable to the magnitude of any 
 draught whatibever. And altJiough this inifrument was 
 defigned for drawing ojdy, yet it may, with a great deal 
 of calc and delight, be applied to the practice of archi¬ 
 tecture, which at the end hereof I lhall dcmonitrate. In the 
 life of tliis inltrument 'tis required to have fuch a te-fquare 
 as is tiled with a drawing table, well known by all archi¬ 
 tects, &c. and the breadth thereof mult be exaCtly equal 
 to i- tire breadth of the inftrument. The lines on thisfeale 
 are of two kinds, viz. parallel, as the lines a, a, a, &c. and 
 centeral, as the lines of tire trigons G H I K L, and the 
 chords over the trigons MGI. The parallelogram C13EF, 
 reprelents of a pillaltcr or column, and C E its femidi- 
 ameter, and confequently cither C D or E F, tlic cente- 
 ral line of the pillafler or column. The diagonal fcalc at 
 the end thereof is treble ; for firfl you have an inch in a 
 lumdred parts, [econdly an inch, and laftly ~ of an inch 
 in the lame proportion. Which fcales are of great ufe in 
 delineating maps, that were meafured with Gunter chain, 
 which is divided into too links. The trigon adjoining there¬ 
 unto hath its bale divided into 90 unequal parts, and is 
 a line of chords from which, to the center, are drawn 
 right lines, wliicJi dhdde the feveral parallels therein in 
 the fame proportion, and thereby you have ro lines of 
 chords to xo different radius, &c. The diagonal lines C F 
 and E D are drawn, to fet thereon the half length of 
 any column or pillafler. The diagonal E D is to be ufed, 
 when any quellion is to be anfvvered by the trigons I K, 
 and the other diagonal CE, when by the trigons MGHL. 
 Tire trigon G hath both its lidcs equal to the femidiameter 
 C E, whereof the outward is divided into go equal parts, 
 each reprcfentiiig a minute, from which are centeral lines 
 4 drawn 
 
p 
 
 'r Sv' 
 
 
 
 
 . C. ■' 
 
 V'. ’f'r’t' /V. ' ' 
 
 
 E.L,-- . I I. .. . ',. .?. -^-:-,-,:----r- -,- 
 
 
 ■ ■ ■ ' 
 
 ... 
 
 I. iJ-L ^ .rx.^_^ ....'tt»nC<»...> -- ! -V 
 
 trv JMtgiiJiyiUT w- ■ ^<4 '■■ ■ ' .w^MMjpf|p«irlU|lCpM||&^k.'«A^O:iV^v * 
 
 - . -; - r - ' 7- -■ — ' ■■ » ■■■ ! y _ ^ xj —^ 
 
 ;V 
 
 
 ■ - T A] 
 
 
 
 V',- 
 
 
 • > 
 
 
 
 i 
 
 i' 
 
 f. 
 
 
 »*< 
 
An Infpectioiicnl Pinin Scale 
 
 Baip' Ji angles o/-' 
 Twickenhajii 
 
 
 
 •^/i//ie€'£it’//niy\a.h\ScaXe^,zj t/^ir ^//^»V^uvv^•ihA. V 
 
 ‘cnpetual iMailit^niiuiciil lui'irvnieiU 
 
 i>/'i/,/{/?i.S/t4i4y{/Wi6‘/i. 47///fA/ /uvc/a 
 _Jlmfs p/7/^e/7l\v^u{r{ca/7^lriy/l/ /ywC’olumii 
 
 i7c7i'7^//ti} o/-^//// 
 
 ~an7£{ 
 
 3 bP 
 


Of the T)ecription and Ufe^ Sac. 
 
 drawn to the center, which divide all parallel right lines 
 drawn therein, in the fame proportion. 
 
 The trigon H hath its fide oppofite to the center divi¬ 
 ded in fucli proportion, as the whole breadth of a piliaitel* 
 is divided by its 8 fillets and 7 flutes; from which divifi- 
 ons are lines drawn to the center, which alfo divide anv 
 line that is parallel to the outlide, in the very lame pro¬ 
 portion. 
 
 The trigons I andK, are divided after the fime manner, 
 as that of I in the divifion of the diameter of a co¬ 
 lumn with flutes only, and that of K with its flutes and 
 fillets. 
 
 The trigons L and M arc made, to furnifli the young 
 ftudent with fcales of all lizes, cither for meafures of feet 
 and inches, as that of L, or feet only, as that of M. 
 
 Problem I. 
 
 The heifji of a pillajier, or column, lehig given, to find 
 the femidiayneier thereof, divided into its^ 30 min. hjf 
 which the whole 'pillafter, or column, with its architrave, 
 freize, and cornifij is me a [tired. 
 
 Let A B be the half height of a column given, to find 
 the femidiameter divided into minutes. 
 
 Pradtice. 
 
 I. Lay 3mur te-fquare on the inhrument, and taking 
 A B in your compalles, move the edge of your fquare to 
 the outlide at C, and at C make a mark on tlic fquare ex- 
 adlly over the line C E, and from that mark fet oil the dif- 
 tance of AB taken in your compalles, on the edge of the 
 fquare as at X. 
 
 Slide back the fquare, ’till the point X lie over the 
 diagonal C F, in the point Y, and then lhal! that edge of 
 the fquare that lies over the trigon G be the femidiameter 
 of the given column, and the centeral lines be divided 
 into 30 min. as required. 
 
 50* Tis belt, when the diameter is thus found, to 
 draw a fine line with a black lead pencil by the fide of 
 the fquare, over the whole breadth of the trigon, and 
 then you may take away the fquare and work from the 
 divifions ol the black lead line, as occafion requires. 
 
 Problem 
 
84. 
 
 Of the Defcription and life 
 
 Problem II. 
 
 The height of a pillafter being given, to find its breadth, 
 and atvifion into 7 flutes and 8 fillets. 
 
 Praftice. 
 
 I. Let it be required to find the breadth and divifion of the 
 pillafter, whofe r height is equal to the aforelaid given line AB. 
 
 z. Place the point X on Y (as before) and then vvili 
 the other edge of the fquare cut thetrigonH in thepoint;?«. 
 
 5. The line nn is the breadth of the pillafter, and its di- 
 vifions, by the centcral lines, are the true breadths of eve¬ 
 ry flute and fillet, as required. 
 
 Problem III, 
 
 The height of a column being given, to find the diame¬ 
 ter, and me afire [or true breadth) of' every flute and 
 /diet, contained in the geometrical upright of the fame. 
 
 The trigon for flutes and fillets is the trigon K, there¬ 
 fore you inuft life the diagonal E D. 
 
 Let the height of the column be (as afore) equal to twice 
 the given line A B. 
 
 T. Place the)- height CX at Z, then will the other edge 
 of the fquare cut the trigon K in the points n n. 
 
 r. The line n n is the diameter, or breadth, of the co¬ 
 lumn, and its divilions by the centeral lines, the true 
 breadths of every flute and fillet, as required. 
 
 Problem IV. 
 
 To find the true meafures of the flutes contained in 
 the geometrical upright of a column, that is fluted without 
 fillets [as often pradiced in the dorich order) to any hedht 
 ajf.gned. 
 
 Let the height of the column be as afbrefaid, equal to 
 tw'ice the given line A B. 
 
 I. Place the point X over Z, then will the other edge 
 of the fquaie, cut the trigon I in the points 0 and 0. 
 
 h The line 0 0 is the diameter of the column, and its 
 divifions, by the centeral lines, are the true breadth of e- 
 very flute, as required. 
 
 That to find the breadth, or magnitude of the 
 flutes and fillets, &c. at the top of the column, where 
 they are narrower than at the bafe, you muft place 
 
 the 
 
of an inflexional phin Scale, &c. 8^ 
 
 tlie diameter upon the refpeaive trigon, fo as to intef- 
 iedl the fides and be parallel to the bafe hereof, and 
 the lines of the trigon will divide the diameter into 
 Its true divifions of flutes and fillets, &c. as required 
 And the like of any other part of the column what- 
 foever. 
 
 Problem VI. 
 
 _ To reduce any part of a line, as a model, minute, &c. 
 into feet and inches, and thereby make this inftriiment 
 univerjal in practice. 
 
 Let a ]}, in the trigon L, reprefent s inches, and tis re- 
 quned immediately to find a fcale of ix inches fuitable to 
 It, whereby any part may be meafured by feet and inches. 
 
 1. Take the line a }> in youi* compalies. 
 
 2. Fix the edge of the fquare to the center H of the 
 tiigon L, and fet off the line a h, on the edge of the 
 fquare from H to i. 
 
 3. Move the fquare towards the line i, r, 3, &c. ’till 
 me point i exadlly lie over the line H y (as on the point 
 E) and draw by the edge of the fquare the line I K, which 
 
 y the I X centeral lines will be divided into ix equal parts, 
 p of which are equal to the given line a h) and is the 
 fcale of IX inches fuitable, or proportionable to the line 
 a b, as required. 
 
 A fecond example. 
 
 Suppofe CE to reprefpnt ao inches, how to £nd a 
 fcale of IX inches proportionable thereunto. 
 
 Pradlice. 
 
 I. The r of xo is 10, therefore lay the edge of the 
 te-fquare to H (as in the lafl example) and fet off the of 
 C E. 
 
 X. Move back the edge of the fquare till the point 
 or r of C E, cut the line H 10. v > 
 
 3. The edge of the fquare being not moved, draw a 
 line by the fame through the trigon L, which by the ix 
 centeral lines will be divided into ix equal parts, repre- 
 fenting inches, and proportional to C E that contains xo 
 inches, as was required to be done. 
 
 (t|=i When your column, or pillaller, contains any num¬ 
 ber of odd inches, radius or diameter, as xy, &c. 
 take i, &c. thereof, as 9, &c, and find the fcale 
 
 Z for 
 
Of the Defcription and Ufe, &c. 
 
 for that number, and that fcale fo found fliall be the 
 fcale proportionable to as required. 
 
 The trigon M is an infinite number of fcales each di¬ 
 vided into tenths, by the lo central lines, as may be at 
 once underftood by a fingle view of the fame. Thefe 
 decimal fcales are of great ufe in meafuring plans of gar¬ 
 dening, and fmall enclofures, taken by foot meafure. And 
 
 The trigon L which is alfo an infinite number of fcales 
 of twelfths, is of great ufe in meafuring plans of build¬ 
 ings taken by feet and inch meafure, which I recommend 
 to the young Undent for the very bell; plain fcale that was 
 ever yet made publick. The excellency and ufe of it 
 will be demonflrated in the feveral parts of this work, as 
 they have relation thereunto. 
 
 Sect. V. 
 
 P L A T E XL 
 
 Of plain Trigonometry. 
 
 I. Of right lined triangles, 
 
 TO Ight lined triangles are diftinguifiied by the difference 
 of their fides, or by the difference of their angles. 
 As to the difference of their fides, they may be all equal, 
 as A, which is called an equilateral triangle, or two fides 
 may be equal and the third unequal, as B, which is cal¬ 
 led an ifofceles triangle, or all the fides may be unequal 
 as C, which is called a fchalenum triangle. And thefe are 
 the diftindtions, in refpedl to their fides. The diftinc- 
 tions of triangles, in refpedl to their angles, are three alfb. 
 
 I. When a triangle hath one angle right, as D, ’tis cal¬ 
 led an orthogonium triangle. 
 
 1. When the triangle hath all the angles acute as A, 
 ’tis called an oxogonium triangle. 
 
 3. When a triangle hath one angle obtufe as C or B, 
 ’tis called an abligonium triangle. And thefe are the dif- 
 tindlions, in refpedl to their angles. 
 
 11 . Of trigonometrical definitions. 
 
 I. Any two fides of a triangle are termed, or called, 
 the fides of that angle. So the fides F G and E G are the 
 fides containing the angle F G E. Every 
 
8 ; 
 
 Of plain Trigonometry. 
 
 z. Every fide of a triangle is the fubtending fide of the 
 angle which is oppofite to it. So in the triangle F G E the 
 fide F G fubtends the angle at E^ and the fide E F fiib- 
 tends the angle at G, and the fide E G the angle at F. 
 For in all plain triangles, the greatefi: fide always fubtends 
 the greatefi: angle, and the lefler fide the lelfer angle, and 
 equal fides equal angles. 
 
 3. The meafure of an angle is an arch of a circle defcri- 
 bed upon the angular point, and is intercepted between 
 the two fides that contain the angle. So the meafure of 
 the angle H I K is the arch c c. See the demonftration 
 of problem II. fedl. I. part II. 
 
 4. Every circle is divided into 3 do degrees, and each 
 degree into do min. See problem XXX. fe« 5 l. II. part. I. 
 
 S- A quadrant is ~ of a circle. See definition id. fedl. I. 
 part. I. 
 
 d. The complement of an arch, lefs than a quadrant, 
 is fo much as an arch wanteth of 90 degrees. So the com¬ 
 plement of the arch C L is H C. 
 
 7. The excefs of an arch greater than a quadrant, is fo 
 many degrees as the arch exceedeth 90 deg. 
 
 8. A femicircle. See defi. id. fedl.I part. I. 
 
 9. The complement of an arch, lels than a lemicircle, 
 to a femicircle, is fo much as the arch wanteth of 180 
 deg. 
 
 10. If a triangle have fbme of its fides equal, it is ei¬ 
 ther equicrural or equilateral. 
 
 11. An equicrural triangle is that which hath two fides 
 equal, and the third unequal. 
 
 iz. An equilateral triangle. See the beginning hereof 
 
 13. A triangle is either right angled, or oblique angled. 
 
 14. A right angled plain triangle is that which hath one 
 right angle and two acute ones. 
 
 I y. An oblique angled plain triangle is that which hath 
 all its angles oblique, viz. one obtule, and two acute. 
 
 id. In all plain triangles, the fum of all the angles are 
 equal to a femicircle, or 180 degrees. 
 
 17. The third angle of any plain triangle is the com¬ 
 plement of the other two, to two right angles, or 180 
 degrees. 
 
 III. Of the conjiruclton of fuch right lines as are applied 
 
 to a circle, for the folution of right lined triangles. 
 
 The right lines applied to a circle for the folution, or 
 calculation of right lined triangles, are chords, fines, tan¬ 
 gents, half tangents, fecants and verfed fines, which may 
 be projedled to any aflign’d radius, as follows. Plate 
 
OJ ■plain Trigonometry. 
 
 Plate XI. 
 
 Geometrically. 
 
 Fig.ii. A ^'hord^ or fubtenfe, is a right line^ joining the ex- 
 
 ti entity of an aich. So AC is the chord of the arch AAIC 
 a. A line of chords, is no more than 90 deg. of the arch 
 of any circle transfer’d from the limb to a right line. 
 
 Conftriidlion. 
 
 I. Draw the right line N V and bifTeft it in O, where¬ 
 on, with the diftance O N, defcribe the femicircle MN V 
 and on O creft the perpendicular O M, and by problem 
 XXX. fedt. II. part I. divide the femicircle into 180 de” 
 1” pbwe one foot of your compafles, and open the 
 
 other firlt to 10 deg. on the femicircle, and defcribe the 
 arch 10, 10, and with the opening V xo the arch xo 
 xo, and the like at every degree; and thereby you’ll trans¬ 
 fer the chords from the quadrant V M, to the diameter 
 or right hue N V, which is the line of chords required. 
 
 3. A right fine is a right line drawn from the end of an 
 aich, perpendicular to the diameter, through to the other 
 end, or tis half the chord of twice the arch. 
 
 Conftruftion. 
 
 I. Fiom 10 deg. on the one fide of the femicircle to 
 10 de^ on the other fide, draw the right line 10 L 10 
 interfetling the perpendicular O M in L. 
 
 X. Perform the like operation throughout the feveral 
 degrees, and thereby you will divide the line O M into the 
 line of fines, as required. 
 
 4. A tangent IS a right line perpendicular to the diame- 
 the extream of the given arch, and termi- 
 nateci by the fecant drawn from the center, thi o’ the cx~ 
 tream ol the faid arch. 
 
 ^^unitructioii. 
 
 On the point V ereft the perpendicular VY, and to it 
 diaw right lines from the center O, thro’ each degree of 
 the quadrant OM V, which lines, fo drawn, fliall divide 
 the peipendicular VY into unequal parts, and lhall be the 
 tangents required. 
 
 T- A 
 
89 
 
 Of plain Trigonometryi 
 
 5. A fecant is a right line drawn from the center thro’ 
 one cxtream of the given arch, ’till it meet with the tan¬ 
 gent, as tlie fecant E do, &c. 
 
 6 . Half tangents are no other than whole tangents, num¬ 
 bered double, as calling 30 min. a whole tangent, and 
 one whole tangent 1 ~ tangents, and therefore deg. of 
 whole tangents is called 90 deg. of ^ tangents, &c. 
 
 7. A verfed fine is a fegment of the diameter, inter¬ 
 cepted between the right fine, and the fine of 90 deg. 
 
 IV. Of divers ajfeEltons incident to plain triangles. 
 
 I. A plain triangle is contained under 3 right lines, and 
 is either right angled or oblique angled. 
 
 X. In all plain triangles two angles being given, the 
 third is alfo given. 
 
 3. In the analyfis of plain triangles, the angles only 
 being given the fides cannot be found but by the realbn 
 or proportion of them. Therefore ’tis wholly requilite 
 that one fide be known. 
 
 4. In a right angle triangled two terms ( befides the 
 right angle) will fuffice to find the third, fo that one of 
 them be a fide. 
 
 S- In oblique angled plain triangles there muft be three 
 terms given to find a fourth. 
 
 6 . In right angled plain triangles there are 7 cafes, and 
 in oblique angled plain triangles s cafes. 
 
 V. Of axioms for the folution of the ix cafet following. 
 
 Axiom I. 
 
 If in a right angled plain triangle, the hypotenufe be 
 made radius, each leg will be the fine of its oppofite 
 angles, but if one leg be made radius, the hypotenufe 
 will be the fecant, and the other leg a tangent thereunto. 
 
 Axiom II. 
 
 In all plain triangles the fides are proportional to the 
 fines of their oppofite angles. 
 
 Axiom III. 
 
 As the fum of the fides of any angle is to their dif¬ 
 ference ; fo is the tangent of half the fum of their oppo¬ 
 fite angles, to the tangent of half their difference. 
 
 A a Axiom 
 
78 
 
 Of plain Trigonometry. 
 
 Axiom IV. 
 
 As the bafe or longeft fide is to the fim of the other 
 fides, lo is the difference of thofe fides to the difference 
 of the fegments of the bale. 
 
 VI. Of the Joint ion of the 7 cafes of right angled 
 plain triangles. 
 
 In right angled plain triangles, I call thofe fides which 
 comprehend the right angle, one the bafe [ptz the long- 
 eft) and the othei the peipendicular, and tlie flope-line, or 
 fide fubtending the right angle, the hypotenufe. 
 
 Cafe I. 
 
 Plate VII. 
 
 The hafe A B 80, and the perpendicular A C do, given 
 to fnd the angle B C A. 
 
 Solution. I. Geometrically. 
 
 I. Delineate BA equal to 80 equal parts of any plain 
 fcale, and on A ereft the perpendicular A C, and make 
 it equal to do equal parts from the fame fcale as you 
 laid down B A. 
 
 1. From the extreams of the bafe at B, and perpen¬ 
 dicular at C, draw the hypotenufe C B. 
 
 3. On C, with do degrees nf 3 line of chords, deferibe 
 the arch a a, and taking the quantity a n, in your com- 
 palfes, and applying it to your line of chords, you will find 
 it to contain '■ 00 the angle required. 
 
 a. B/ Trigonometrical calculation. 
 
 Analogies. 
 
 Fir ft, as the perpendicular C A do, is to the bafe B A 
 80, fo is the tangent of 45- degrees, to the tangent of 37 
 deg. whofe complement to a quadrant, or 90 deg. is 5-? 
 deg. the angle required. Or, 
 
 Secondly , As B A 80 is, to the tangent of 45- deg. fo is 
 A C do to the tangent of 37 deg. whofe complement 
 is yg deg. the angle required. 
 
 Cafe II. 
 
 The haje A B 80, and the angle B C A, si deg. given, 
 to find the perpendicular AC. 
 
 Solution. 
 
Of plain Trigonometry. 
 
 Solution. I. Geometrically. 
 
 I. Delineate the bafe A B and make it equal to 8o 
 and on A ereft the perpendicular AD. ’ 
 
 a. Make the arch n m, equal to the complement of 
 me given angle, and through m draw the hypotenufe 
 B C, which will interfedt the perpendicular in C. 
 
 ^ C A, being laid on your Icale of equal 
 
 rS”^^' requi- 
 
 1. By Trigonometrical calculation. 
 
 Analogy. 
 
 K ^ ® ^ ^ ^3 deg. is to the 
 
 1 complement, of the 
 
 angle B G A 3 7 deg. to the perpendicular C A do as re¬ 
 quired. ^ 
 
 Cafe III. 
 
 The hypotenufe B G 100, and the hafe B A 80 viven 
 to Jind the angle B C A. * 
 
 Solution. I. Geometrically. 
 
 I. Make BA equal to 80, and on A erett the perpendi- 
 cuDr A D, and on B, with the length of the hypotenufe P 
 
 B C, delcribe the arch 0 m, inrerfedting the perpendicular 
 in C. i i 
 
 %. On C, with do deg. of your line of chords, deferibe 
 the arch r s, and take the arch r s, and meafiire it on 
 your line of chords, and it will contain si deg. the 
 angle required. 
 
 %. By Trigonometrical calculation. 
 
 As the hypotenufe B C 100, is to the radius, or fine 
 of 90 deg. fo is the bafe BA 80, to the fine of si deg 
 the angle required. 
 
 Or thus. > 
 
 As B A 80, is to B C 100, fo is the radius to the 
 fine of the complement of si deg. the angle re¬ 
 quired. 
 
 Cafe IV. 
 
 The hypotenufe BC 100, and the angle B C A T 3 deg. 
 given, to find the hafe. 
 
 ' Solution. 
 
 79 
 
 rig. Ill, 
 
 ig. IV. 
 
9^ 
 
 rig.V. 
 
 Fig. vr. 
 
 Of plain Trigmometry. 
 
 Solution. I. Geometrically. 
 
 I. Draw AC at pleafure^ and on with 6o deg. of your 
 line of chords, defcribe the arch nn, and make the an¬ 
 gle C equal to 5-3 deg. the angle given, and draw the 
 hypotenufeBC equal to 100. 
 
 1. On B, with 60 deg. of chords, deferibe the arch 0 0, 
 and make the angle B equal to the complement of C, and 
 draw BA, which lhall cut C A, the perpendicular in A, and 
 being meafured on your fcale of equal parts, will contain 
 80, the length required. 
 
 a. By Trigonometrical calculation. 
 
 As the radius, or fine of 90 deg. is to the hypotenufe 
 B C 100, fo is the fine of the angle B C A 5-3 deg. 
 to the bafe 80, as required. 
 
 Or thus. 
 
 As the radius, or fine 90 deg. is to the fine of the an¬ 
 gle BCA ^3 deg. fo is B C the hypotenufe 100, to 
 the bafe 80, as required. 
 
 Cafe V. 
 
 The angle ABC 37 cleg, and the perpendicular AC bo 
 given, to find the hypotenufe B C. 
 
 Solution. I. Geometrically 
 
 I. Draw A B at pleafure, and on A eredl the perpendi¬ 
 cular A C equal to bo, and on C, with bo deg. of chords, 
 deferibe the arch i i, and make the angle C equal to 
 5-3 deg. the complement of the given angle, and 
 draw the hypotenufe C i B, which will interfefl; the bafe 
 B A, in B, and being meafured on your fcale of equal 
 parts will contain 100, as required. 
 
 a. By Trigonometrical calculation. 
 
 As the fine of the angle ABC 37 deg. is to the 
 perpendicular A C bo, fo is the radius, or fine of 90 deg. 
 to B C the hypotenufe 100, as required. 
 
 Cafe VI. 
 
 The^ hypotenufe BC 100, and the perpendicular AC 
 bo, given, to find the hafe B A. 
 
 ..!■ 3 
 
 Solution. 
 
Of plain Trigonometry. ^ 
 
 Solution. I Geometrically. 
 
 I. Draw B A at pleafure, and on A eredl the perpen- ^ig- vir. 
 dicular A C, and make it equal to 6o, and on C, with 
 the length of the hypotenufe^ defcribe the arch Z' in¬ 
 terfering the bafe in B. 
 
 X. The length B A is the bafe required. 
 
 a. Bj Trigonometrical calculation. 
 
 Analogy. 
 
 As the hypotenufe B C loo, is to the radius^ or line of 
 90 deg. fo is the perpendicular A C 60, to the line of 37 
 deg. the angle ABC. Then 
 
 As the radius is to the hypotenufe loo^ fo is the co- 
 fme of the angle ABC 73 deg. to B A 80, the bafe 
 required. 
 
 Or thus. 
 
 Multiply the perpendicular into it felf as allb the hy¬ 
 potenufe, and fubllraftthe lelfer produfl: from the greater, 
 then lliall the fquare root of the remainder be the length 
 of the bafe required. 
 
 Cafe VII. 
 
 The bafe BA 80, and the perpendicular A C do given, 
 to find the hypotenufe. 
 
 Solution. I Geometrically. 
 
 I. Make A B equal to 80, and on A eredl the perpen- Fig vnr. 
 dicular A C, equal to do. 
 
 a. From B to C draw the hypotenule required. 
 
 X. By Trigonometrical calculation. 
 
 Analogies. 
 
 As the perpendicular A C do is to the tangent of 47 
 deg. fo is B A the bafe 80 to the tangent of 37 deg. 
 the angle ABC. Then 
 
 As the cotangent of 37 deg. is to the bafe B A 
 80, fo is the tangent of 47 deg. to the hypotenufe B C 
 100, required. 
 
 Or thus. 
 
 By theorem V. fe6l. III. part. I. multiply the bafe into 
 its felf^ as alfo the perpendicular, and add both the pro- 
 4 B b duds 
 

 Of plain Trigonometry. 
 
 '■ 
 
 du( 5 ts together, then fliall the Icjiiare root of that Him, 
 be the hypotenufe required. 
 
 VII. Of the Joliition of the j cafes of ohliqiie angled 
 plain triangles. 
 
 Cafe I. 
 
 The Jides B C 5-0 and C A do with the angle ABC 
 T? deg. given, to find the angle C A B. 
 
 Solution. I Geometrically. 
 
 Fig. IX 
 
 I. Draw B A at pleafurc, and make the angle ABC 
 equal to the given angle, and make B C equal to yo. 
 
 a. On C, with the dillance of C A do, deferibe the 
 arch t t, interfecling the bafe in A, whereon, witli do 
 deg. ol diords, dclcribe the arch am, which being mea- 
 lured on the line of chords, will be equal to ix deg. 50 
 min. the angle required. 
 
 X. By Trigonometrical calculation. 
 
 Analogies. By axiom II. 
 
 As the fide C A do, is to the fine of the angle ABC, 
 X7 deg. fo is the fide BC yo, to the fine of ixdeg. 30 
 min. the angle required. 
 
 Or thus. 
 
 As the fide B C yo, is to the fide C A do, fo is the 
 fine of x- deg. the angle A B C, to xx deg. 30111111. the 
 angle required. 
 
 Cafe II. 
 
 The fides B C yo, and C A do, with the angle A C B 
 131 deg. 10 mm. given, to find the other angles CAB 
 and A B C. 
 
 Solution. I. Geometrically. 
 
 I. Delineate B C and C A, making the angle C equal 
 to the given angle. 
 
 X. Join A B the bafe, and with do deg. of the line 
 of chords on the points A and B, defenbe the arches 
 0 p and cq r, which being leverally meafured on the line 
 of chords, will be the quantity of the angles required. 
 
 X. By Trigonometrical calculation. 
 
 Analogic. By axiom III. 
 
 As the full! of the fides B C and C A 100, is to their 
 
 difference. 
 
Of plain Trigonometry. 
 
 difference, viz. lo. fo is the tangent of ~ the oppofite 
 angles 14. deg. 43- min. to tlie tangent of their difter- 
 ence, viz. % deg. 4}- min. Then 
 
 This ^ difference fubftrafled from the ^ fum of the op¬ 
 pofite angles gives the inferior angle ; but added to 
 the ~ fum of the oppofite angles gives the liiperior an¬ 
 gle. 
 
 Or thus. 
 
 As the I fum of the given fides is to their t difte- 
 rence, fo is the tangent of ^ the oppofite angles to tlie 
 tangent of their difference. 
 
 Then add and fubllratt as before diredled. 
 
 Cafe III. 
 
 The angle ABC and CAB, with the fide B C jo, 
 oppofite to the angle CAB given, to find the fde or 
 haje AC. 
 
 Solution. I. Geometrically. 
 
 I. Make B C equal to 50, and make the angle C B A 
 equal to 131 deg. 3 o min. as given, and di'aw B A infinitely. 
 
 r. Make the angle B C A equal to the complement of 
 the two given angles, to 180 deg and draw A C, which 
 will interfedl B A in A. 
 
 3. The line A C is the bafe required, and if mcafured 
 on your plain fcale, will be equal to 100 of thofe parts, 
 as B C contains yo. 
 
 2. By Trigonometrical calculation. 
 
 Analogy. By axiom II. 
 
 As the fine of the angle C AB 27 deg. is to the fide BC 
 SO, fo is the fine complement of the angle ABC, viz. 
 49 deg. to 30 min. the bafe 100. 
 
 Cafe IV. 
 
 The fides B C 60, and C A 100, with the angle C 22 deg. 
 30 min. comprehended hy them given, to find the fdeAB. 
 
 Solution. I. Geometrically. 
 
 I. Make AC, B C, and the angle C, equal to the mea- 
 fures given, and from tlieir extreams draw A B, and it 
 ftiall be the fide required. 
 
 2, By 
 
 95 
 
 Fio. XI, 
 
 Fig. Xli. 
 
 
 ; ' 4=1 
 
 
 
 f-t 
 
X. By Trigonometrical calculation. 
 
 I. Find the angle at A, by axiom I. or II. then tlie 
 analogy is thus, as the fine of the angle C A B is to B C ; 
 fo is the fine of the angle A C B to the fide AB, required. 
 
 Cafe V. 
 
 The 5 fides AC loo, A B yo, BC given, to find 
 the angle B C A, or the angle CAB. 
 
 Solution. I. Geometrically. 
 
 I. By problem XIV. fedl. II. part I. delineate the tri- 
 I'ig. xm. angle ABC equal to the 3 given fides. 
 
 X. With do deg. of chords on A and C, defcribe the ar¬ 
 ches nn and rr, which being meafurcd on the line of 
 chords, will Ihew the quantity of each angle as required. 
 
 X. By Trigonometrical calculation. 
 
 For the folution of this cafe, two operations are requi¬ 
 red, viz. the lirlt, to find the fegraent of the bafc A D 
 and DC, and the fecond to find the angles required. 
 
 Analogy. Firft by axiom IV. 
 
 As the fum of the bafe A C 100, is to the fides A B and 
 BC no, fo is the diflerence of A B and B C, {viz. 10) 
 to the dift'erence of the fegments of the bafe {viz. 
 14..) This fegment added to the bafe, viz. 100, the 
 film is 114., whofe 7, viz. yy, is the fuperiour feg¬ 
 ment D C, which being fubftrafted from A C 100, leaves 
 A D 43 the lefier fegment. And now there are confti- 
 tuted two right angled plain triangles, by which the an¬ 
 gles may thus be lound, by this analogy of cafe III. of 
 right angled plain triangles. 
 
 Analogy. 
 
 As the hypotenufe BC do, is to the radius, or fine of 90 
 deg. fo is the greater fegment D C yy, to the angle D B C, 
 w hofe complement to 90 deg. is the angle BCA, required. 
 
 Again, 
 
 As the hypotenufe B C do, is to the radius, lo is the 
 lefier fegment AD 43, to the angle DBA, whofe comple¬ 
 ment to 90 deg. is the angle BAG, required I 
 
91 
 
 Of -plain Trigonometry. 
 
 I advife that the young ftudent be peifed in thefe ix 
 cafes, before he proceed any further. For hereon depend 
 not only the principles of framing all kinds of roofs of 
 buildings, meafuring all kinds of heights and diftances, 
 accelfible, or inacceffible, furveying of land, meafuring" 
 &c. but alfo of navigation, fortification, and gunnery, 
 which fome youths may delight in the ftudy of; belides 
 the fubjedls hereof, they being both profitable and de- 
 lightful. 
 
 Sect. VI. 
 
 Of the Geometrical Conjiru^ion of Draughts, Tlans 
 and Maps of Lands, Gardens, Farms, Build¬ 
 ings, &c. 
 
 enumerate the many mathematical inftruments 
 invented for this purpofe, and to defcribe their ufe 
 would be but a needlefs amufing work, feeing thac herein 
 the only inftruments that I make ufe of are the comnron 
 Gunters chain, and a common five and ten foot rod, divi¬ 
 ded into feet and inches, &c. with which I firall fhew how 
 to dcftiibe any plan, with much left trouble and in much 
 left time than by the help of any theodilite, plain table, 
 chcumferentor, &c. and, if I miftake not, far more exadt. 
 
 When the meafure of any length is taken by a chain, 
 and contains lo chains 75 links, ’tis thus written 10 : 75^ 
 fo alfo is 73 chains 4 links, thus 73 ; 04; and when any 
 length is meafured that is left than a chain’s length, as 7 3 
 links, ’tis either written thus 00 : 73, or thus : . 73, with 
 a period or full ftop before it, in the ftme manner, as a 
 decimal ffadlion. 
 
 The reafon why the links are often times exprefs’d 
 according to the laft way, is, becaufe in taking the mea- 
 fures of offsets, (which are very often under the length 
 of a chain) there is no room to exprefs them according 
 to the former way, between the offsets taken. See the 
 meafures of the offsets taken from the line LK, fig. 
 XXL plate XL where may be feen, at one view, a de- 
 monftration thereof. 
 
 C c 
 
 The 
 
Of the Geometrical Conjiruflion of Draughts, &c. 
 
 The appurtenances belonging to the chain are ten linall 
 rods^ each about two loot in length, lltod and llrarp- 
 pointed with iron, to flick in the ground at the end of e- 
 very chain’s length, when a length is meafuring. 
 
 The manner of meafuring a length is as follows. 
 
 1. One man takes an end of the chain in his hand, and 
 walks towards the place he is to meafure to, taking with 
 him, under his arm, the aforefaid ten flicks. 
 
 2. When he has walk'd the length of the chain, the 
 hindermoll man caufes him to move either to the rigltt 
 hand or to the left, &c. ’till he has placed him in a right 
 line pofition from him, to the place to which they mea- 
 llire. Which being done, and the chain laid very llreight 
 and tight, the foremoft man flicks down one of his flicks 
 and leaves it, and then walks oji forward towards the mark 
 he meafures to, ’till the hindermoft man comes up to 
 the flick, the firfl flick’d down. And then (as aforefaid) the 
 hindermoft man directs the foremoft in a right line with 
 the mark. Where after laying tite chain flreight, he flicks 
 down a fecond flick, and then walks forward towards the 
 mark, and the perfon behind, alfo, bringing thofe flicks 
 with him, as he takes them up at the end of every chain, 
 ’till he comes to the next, and there repeats the fame a- 
 gain, &c. and thereby he knows what number of chains 
 is contain d in that length fo meafured. 
 
 Problem I. 
 
 Plate XI. 
 
 Let it he required to make a plan of the field B DE 
 EG HIKE, fig. XXL plate XL 
 
 1. Walk round the bounds of the fame, and at every 
 fudden turn ereft a flick, or flafl) of about five foot high, 
 with a piece of white paper on the top of each, as at the 
 feveral turns B, D, E, F, G, H, I, K, L. 
 
 2. Go into feme convenient part of the field, from 
 which you may fee all the flation flaffs before eredled, as 
 at A, and there drive down a fmall flake. 
 
 3. Meafure from A to any one of the flation flaffs, as 
 to L, and note down the chains and links contain’d there¬ 
 in, and alfo meafure from L to K, and from K to A, and 
 then you have the lengths of the three fides of the trian¬ 
 gle ALK, given. 
 
 4. By problem XIV. fe£b. I. part 1 . make (or deferibe) 
 the triangle ALK, whofe three fides fliall be equal to the 
 aforefaid three lines meafured. 
 
 3 
 
 y. Meafure 
 
Oj the Geometrical Confirumon of Draughts, 8cc. 
 
 S- Adeafnre and delineate the feveral offsets m o, m o &c. 
 by problem VI. feft. I. part 11 . and deferibe the crooked 
 line L, o, o, o, o, o, o, K. 
 
 6 . Meafure from the ftation Half at K to I, and alfo 
 from I to A, and then fuppofing A K to be a third given 
 line, by pioblem XIV. fedl. I. parti, deferibe the triangle 
 A K I, and from the line KI fet off the feveral offsets m r, 
 mr, mr, &c, as aforefaid, and delineate the crooked line 
 K r r r. See. I. 
 
 Aleafmc from the ftation ffaff at I, to the next atH, 
 and meafure the diftance IH, and from H to A, and fup- 
 poliiig the line A I to be a third line, by problem XIV. 
 lea. I. part I. delineate the lines I H and H A, and by 
 problem VI. fedl.I. part II. meafure and fet off the feveral 
 offsets m s, ?n s, 7n s, &c. and trace the crooked line 
 I r r r r, &c. H. 
 
 8. Proceed in the very fame manner from H to G, and 
 from thence to F, E, D, B and L, and thereby you will, 
 with gieat eafe, exadlly deferibe the plan, or figure of the 
 field, as required. 
 
 ^Vhen you have feveral fields to furvey, then you muff 
 know how to place your ftation in the fecond field, af¬ 
 ter you have completed the firft, which is to be perfor¬ 
 med as follows. 
 
 I. Go imo the fecond field, fig. XXII. and place your 
 ftation ftaffs in convenient places about the fame, as at 
 N, 0 ,P,CL,U.R, T and V. 
 
 a. In a convenient place, as at M, fix your ftation as 
 you did in the former field at A. 
 
 3. Meafure from A to M and fet down the meafure, 
 and alfo meafure from L to M, and note that alfo. And 
 then you have the length of two given lines. And if you 
 fuppofe A L to be the third, then by problem XIV. 
 
 I. part. I. deferibe the lines AM and LM, interfeding 
 each other in the point M, from which you may meafure 
 to every ftation ftaff, &c. and form that field, in the very 
 fame manner as the firft, and the like rule from thence 
 to X, fig. XXIII. and from that to others, &c. 
 
 N. B. When any inclofure is fo fituated, that you can¬ 
 not go within fide to make a plan thereof, as in the 
 preceding, then you muft go round the fame without- 
 fide, and deferibe the plan thereof as following. 
 
 99 
 
 Fig. XX[ 
 
 I. Make 
 
oo 
 
 Of the Geometrical Conjiruclion of Draughts, &c. 
 
 I. Make an cyc-draught thereof (which is a rough 
 draught on paper) wherein delcribe every inditddual an¬ 
 gle turning, &c. 
 
 1. Standing at any part thereof, as at in, conceive the 
 line mAB, and from it meafure the fcveral oftsets /, /, I, 
 &c. and at their extreains draw the fide of the field FED, 
 &:c. according to problem feft. I. part II. 
 
 5. Standing at m, conceive the line mg h, and by prob. 
 Y. fcCl. I. part II. meafure the angle m, and note it down, 
 and afterwards take the fcveral offsets 0, 0, 0, &c. as be¬ 
 fore, and then place yoiirfelf in another convenient place, 
 as at g h, and tliere conceive the line h i, gh, and then 
 proceed as before, and fo from thence to other flations, 
 till you have taken the whole circumference of the field ; 
 after which delineate the fime from the eye-draught, by 
 the rules before laid down, and thereby you will have an 
 exadt plan, as required, notwithttanding you were not 
 admitted within the fame; which oftentimes happens by 
 wood, water, &c. or when the land is a perfon’s who will 
 not allotv a furveyor to go thereon. 
 
 If you conceive the afbrefaid figure (or at leaf! the cir¬ 
 cumference thereof) to be the fide of fo many llrects as in- 
 clofes that quantity of ground, you may, by the very 
 fame rule, delineate any parifli, town, city, &c. provided 
 that as you go on, you meafure the offsets to the right 
 hand fide of the flreet as well as to the left, as the offsets 
 m r, in r, &c. in the figure, and by their extreains delcribe 
 that fide of the flreet, &c. alfo. But when the fides of 
 ftreets are flreight lines, then there need none of thefe 
 offsets, and the work is performed with much lefs trouble, 
 and therefore I made choice of this mofl difficult part for 
 an example. 
 
 Pro blem II. 
 
 Plate XII. 
 
 To make a plan or draught oj any garden, wildernefs, 
 e^c. and keautify the fame with proper colours. 
 
 Let I K L M be a garden divided into walks, parterres, 
 borders, &c. and ’tis required to draw a draught of the 
 fame, and to diflinguifli each particular with proper co¬ 
 lours. 
 
 I. Draw the centeral line O N. 
 
 a. Meafure the breadth of the middle walk B, and at 
 the parallel diltance of r B C, draw the lines B F and C G, 
 infinitely. 
 
 3. On 
 
I 
 

lOl 
 
 Of the Geometrical Conjlruciion of Draughts, See. 
 
 5. On any point of the centeral line as at O, draw the 
 line P CL, at right angles infinitely, and at the parallel 
 diftance of P V (the breadth of the terrace) draw the line 
 V K infinitely. 
 
 4. At the parallel diftance of V O, draw the line O P, 
 and divide that parallel diftance with two other parallel 
 lines in fuch proportion as the Hope and verges are divided. 
 
 f. At the diftance of O A draw A D infinitely, as alii) 
 the line E PI at the parallel diftance of A E, and likewile 
 the lines XX and YY, at fuch diftances as their breadths 
 contain. And thus have you, by thofe parallel lines, di¬ 
 vided the feveral crofs walks, &c. therein, in refped: to 
 their breadths. And to find their terminations, or inter- 
 fedtions, proceed as follows. 
 
 I. On the lines A D and E H, from the points B C and xxiv 
 F G, fet oft' the meafures BA, CD, E F, G H, and draw 
 the lines A E and PI D. 
 
 1. The two parterres being thus enclofcd, and their fe¬ 
 veral parts being all parallel to each other, therefore mca- 
 fure the diftances between the feveral lines contained 
 therein, and draw every particular line parallel to the 
 centeral line N O. 
 
 3. Give to every right line its particular length, and 
 deferibe every circular line by the rules laid down in 
 fedl. II. part I, and thereby you will complete the Icvcral 
 parts therein contained. And as the other outer walks, 
 
 Hopes, verges, &c. are all parallel to the aforefaid, there¬ 
 fore at thole parallel diftance.s, deferibe every line, and 
 thereby you will complete the whole draught, as rccpiired. 
 
 And what is faid of the delineation of this, the fame is to 
 be underftood of all others of the like nature. And, indeed, 
 he that is well acquainted with all the preceding pro¬ 
 blems, is enabled to make a plan of this or any other gar¬ 
 den without any more diredfions. And therefore it being 
 needlefs to treat any further thereon, I fiiall leave the inge¬ 
 nious ftudent to the praftice thereof And for his exercife 
 I have fubjoin’d the plan of a wildernefs, fig. XXV plate 
 XIII. wherein arc contain’d fome few artinatural lines, that 
 may be worthy of his confideration, and not a linall help 
 to invention in defigning gardens after that rural manner; 
 which are not entirely new, but far preferable to the moft 
 regular fet forms hitherto pradlifed (as I obferv’d before) 
 in moft paits of England, to the great difadvantage of the 
 proprietors, and lhame of the pretended performers. And 
 to demonftrate more plainly, that the laying out of .gar¬ 
 dens has no fort of recourfe to the exterior figure, or 
 
 D d bounds 
 
102 
 
 Of the Geometrical Conjiru&ion of Draughts, See. 
 
 bounds thereof, being regular, I have fubjoined the plan, 
 fig. XXVI. plate XII. wherein is delineated an artinatural 
 walk, which demonftrates that the moll beautiful gardens 
 are to be made in the molt irregular forms or boundaries. 
 Altho’ the practice hitherto has been the reverfe. And 
 thereby oftentimes to make a garden regular {or raiher 
 totally ruin it) the gentleman has been advifed to pur- 
 chafe a part of his neighbours land at a very dear rate, 
 purely for tltc lake of regularity, which in all gardens 
 Ihould be avoided, as may be feen in plate XIV. which 
 is an entire garden according to the truth of defigning, 
 wherein you may behold art and nature in conjunction 
 w ith each other, which in gardening is a general axiom to 
 be obferved, &c. 
 
 K B. To reprefent grafs you mult ufe fap-green, and 
 gumbougc lightifd for land or gravel. 
 
 Problem III. 
 
 Plate XV. 
 
 How to make the map of any eflate, farm, lordfotp, 
 mannor, &c. Let it he required to make a map of the 
 lands, fg. XXVII. 
 
 I. By problem 7. feCt. I. part II. make a plan of the 
 dwelling houfe B, and liable A. 
 
 1. By problem p. felt. I. part II. take the quantity of 
 Fig.xxvir. the angle P, and draw the line F G. 
 
 3. Alfo take the quantity of the angle G, and draw the 
 line G, e, equal to the meafure taken, and by problem VII. 
 left. I. part 11 . make the plan of the barn D and the liable E. 
 
 4. Take the angle H, and draw H I equal to its meafure. 
 p. Take the angle I, and draw I N equal to its meafure. 
 
 6 . Meallire the lines I, 0, 0, 0, and 0, N, and complete 
 the trapezium I, 0, 0, N. So will 0, 0, be one end of the 
 barn D E. 
 
 7. Meafure either or both of the angles, 0, and 0, and 
 complete the oblong plan of the barn D E, and alfo by 
 the fame rule complete the barn D I. 
 
 8. Alcafure the lines N K and L K, and delineate them. 
 So lliall you have completed the houfe, barns, Ilables, 
 yards, &c. 
 
 9. In tlie field Z, in any convenient 'part, as at Z, drive 
 down a Hake lor a Hation. 
 
 10. Meafure the lines V Z and T Z, and delineate 
 them by the latter part of problem I. hereof 
 
 II. Your 
 

 # A r0^ 
 
 f^i n ?j 
 
 fJ^ 
 

IS' i’fi' 
 
 S' f f f f f J*« 
 
 y ¥ f ¥ M 
 
 
of the Geometrical Gonjim&ion of Draughts, See. 
 
 II. Your ftation point being thus placed, meafure from 
 thence into every angle, or to Inch hidden turns in tlie 
 hedges, as are remarkable, and then proceed in all relpetls 
 as is laid down in problem I. hereof, and thereby you will 
 exailly delineate a true map of the firm, as rec|uired. 
 
 Advertifement. 
 
 I do hereby advife every gentleman, that when they 
 imploy a land furveyor to meafure and map an eltate, 
 they caufe him to deferibe every timber tree contain’d 
 therein, and that the timber of every tree be meafured, 
 and the quantity of that meafure written underneath each 
 particular tree, with the firft letter of the tree's name, 
 as an E for elm, an O for oak, an A for ahi, &c. and 
 thereby the true value of an eftate, both of land and 
 timber, may be known at all times, without any fort of 
 trouble. Alfo if a gentleman lets any land by leafe, or 
 otherwife, ’tis not in the power of his tenant to wrong 
 him of any one tree of timber, contained in the lands to 
 him demifed, and many other excellent advantages too 
 tedious here to mention. 
 
 Note, That when timber-trees Hand fo very thick fas 
 in a wood) that the reprefentation of every tree, with 
 its meafure, cannot be inierted, then at all fuch times 
 the furveyor mult reprefent the bafts of every tree, 
 by a point or period, with numerical figures to each, 
 as I, a, 3, 4., y, &c. Which numbers refer you to 
 the very lame numbers in a column placed on one 
 fide of the map, againll which Hands the true folk! 
 content of every tree, as each point, or figure repre- 
 fents. See LM of Nuns wood, with its tables of quan¬ 
 tities, &c. 
 
 Problem IV. 
 
 Plate XVI. 
 
 How to increafe or decreafe any draught, at pleafure. 
 
 Let abode f g h i k I m n 0 p, &c. be a fmall map 
 of a farm, and ’tis required to increafe the fame four 
 times its magnitude. 
 
 I. In any part of the fame, as at A, make a point, and 
 from that point, through all the feveral angles, draw right 
 lines, and continue them infinitely. 
 
 r. Open your compalles from A, the given point, to b, 
 and on the fame line let that dillance from b to B. 
 
 4 3. Make 
 
iOi|. Of the Geometrical Conjiruftion of Draughts, ike. 
 
 3. Make C c equal to A c, and draw the line B C. 
 
 4. Make D d equal to Ad, and draw the line CD, and 
 in the fame manner, proceed ’till you have pafs'd throuo^h 
 the whole, and thereby you’ll encrcafe the map, as re¬ 
 quired. 
 
 Note, That by doubling the diftance from the uiven 
 pennt A to the feveral angles, you thereby (as afore- 
 laid) encreafe the figure four times ; therefore the 
 double of that is eight times, and its half but two 
 times, and confequently its quarter but once. So that 
 from thefe proportions you may increafe, or decreafe, 
 any map to any proportion, as may be required. 
 
 Problem k . 
 
 Plate XVI. 
 
 How to defer the [and accomit for) the dhnmution of 
 the breadths of long walks, avenues, v'ljids, &c. 
 
 'Tis obfervable, that the breadth of long walks, ave- 
 Fig. XXX ‘'•Ppe^iFS to be much narrower at the further 
 
 ■ end, than at that end where the perfon hands, notwitlt- 
 ftanding the fides of the walk are adlually parallel to each 
 other. But what is the reafoii thereof, no gardener, or 
 indeed any other, has yet accounted lor it to the publick. 
 It is occafion’d as follows, 
 
 I. vVll objecls that appear equal in height, or breadth, 
 are feen under equal angles; but when objedls appear un¬ 
 equal, and aie equal. Inch objefts are feen under unequal 
 angles. 
 
 Demonflration. 
 
 I. Let the lines CE G and ADF, reprefent the hedge 
 lines of a walk, &c. 
 
 r. Draw the centeral line B Z, and let the points C, E, 
 G, E, p. A, be remarkable places in the hedge lines CEG 
 and P DA, and let the line W X V be drawn at right an¬ 
 gles to Z B, as alfo the lines D E and AC. 
 
 5. Make TB equal to TX, and draw the lines CB and 
 A B, as alfo W T and VT, as alfo E B and D B, and G B 
 and F B. 
 
 4. The objedls C A and 'V W, being equal to each other, 
 and they lying placed at equal diftances therefrom, as at T 
 and B, are feen under equal angles, and do appear to be 
 equal to each other ; but the objeiTs D E and F G, ap¬ 
 pear both lefs than VW or AC, by reafonthe angle EBD 
 
 and 
 
io6 Of the Geometrical ConJlruBion of Draughts^ dec. 
 
 rijxxxH. 
 
 P R O r. L E M VII. 
 
 Hovi-f to give the exa& height to any Jlatue placed 071 a 
 liiilding, that the fame Jhall appear equal to the coy 7 mion 
 height of a rnari Jiaridhig on the ground. 
 
 Let D C be the common lieight of a man (as five foot 
 nine, or ten inches) and ’tis required to place a llatue on 
 the building at L flrall that appear equal in height there¬ 
 unto. 
 
 I. At any convenient place, as at A, upon the level of 
 tlie building, place your Ilation, and draw the lines AC, 
 A D and A 1 . 
 
 a. On the point A, with any diftance, deferibe the arch 
 B K F(i, and make F G equal to E B, and draw the line 
 iV G FI. 
 
 3. Continue Cl to H, fb lhall I H be the height of the 
 objccl, required. 
 
 Demonllration. 
 
 The angle E A B is equal to G A F, therefore fmee 
 H 1 is feen under the very fame angle, as D C, by problem 
 V. hereofi H 1 is in appearance equal to DC, and is the 
 true height of that objeft, which is what was to be de- 
 numllrated. 
 a 
 
 T H. E 
 
THE 
 
 PRACTICE 
 
 O F 
 
 Architeclure, Gardening, Menfuration, and 
 
 Land-Surveying, Geometrically demonftrated. 
 
 PART III. 
 
 Of Geometrical Axioms and Analogies, for the 
 Menfuration of all Kind ol Lines, fuperfici- 
 al Figures, and folid Bodies, ^c. 
 
 Since tie Menfuration of all Kind of Work is 
 for the Generality perform'd by Crofs Multiplication, 
 therejore I 'will firjl explain the fame, and after¬ 
 wards proceed to Menfuration in general. 
 
 parts. 
 
 Sect. I. 
 
 Of Crofs Multiplication, 
 
 E T it be required to multiply feven feet, three 
 inches, fix parts, by five feet, four inches, fix 
 
 3 
 
 Place 
 
io8 
 
 Of Crofs Multiplication. 
 
 Place the numbers thus 
 
 feet.inch.part. 
 ; 7 : 3 : 
 
 [ J- ; 4 : (> 
 
 I. Multiply y feet by and the produft is 3 7 
 
 r. Multiply the feet into the inches, as s] 
 into 5, which is ly, ix of which make or 
 foot, which place under the feet, and the( 
 remaining 5 under the inches. 
 
 3. Multiply the 7 feet into 4 inches, and. 
 the produdl is x8, wherein is twice ix, andv 
 4 remaining, which is x feet and 4 inches,( 
 which place under feet and inches. 
 
 4. Multiply the inches into themfelvesl 
 
 and their produfl; is ix, which is i inch,; 
 which place under inches. j 
 
 y. Multiply the parts into the feet, as 
 times 7 is 4X, wherein ix is contain’d/ 
 thrice, and 6 remaining, which is 3 inches) 
 and 6 parts, which write under inches and> 
 parts. 
 
 6 . Multiply the 6 parts into the y feet,) 
 
 and the product is 30, or x inches; which) 
 alfo write down. J 
 
 7. Multiply the parts into the inches, as^ 
 
 6 into 3, and the fum is 18 parts, or i partv 
 and ;, which write down under parts, as( 
 thus. 
 
 8. Multiply the 6 into 4, and the proO 
 
 dutt is 14, or X part.s, which alfo place under) 
 parts, thus. } 
 
 Lajily, Multiply the parts into them-^ 
 felves, and their produed; is 3d, of which) 
 144 make 1 part, therefore 36 is 
 
 And the fum is 
 
 I f 
 
 3 I- 
 
 S E C T. 
 
lO^ 
 
 Sect. II. 
 
 Of Geometrical Axioms for the Menfuration of Lines 
 and fuprficial figures. 
 
 Plate XVII, 
 
 Problem I. 
 
 To meafure the geometrical fquare ABCD, whoje fides 
 are each equal to i 6 foot 6 inches. 
 
 Rule. 
 
 Multiply any one fide, as AB, id foot 6 inches, by any 
 other of the fides, as B D, and the produdl will be ^7^ ^ig- i. 
 foot 16 inches, the fuperficial content required. 
 
 Problem II. 
 
 To meajure the parallelogram ABCD, whofe longejl 
 fide is equal to z8 foot 9 inches, and its fsorteji to 6 foot 
 6 inches. 
 
 Rule. 
 
 Multiply the length z8 foot 9 inches, by the breadth 
 6 foot 6 inches, and the produft is 186 the fuperficial 
 content required. 
 
 Problem III. 
 
 To me afure the triangle N M O, whofe longefl fide N O 
 is equal to 4.x foot, and the perpendicular MA to id foot. 
 
 ^ That before any right lined triangle is meafured, a 
 perpendicular line is let fill upon the longeffc fide (or 
 bafe) from the oppofite angle. 
 
 Rule. 
 
 Multiply half the perpendicular (viz. 8.) by 42, the 
 length of the bafe N O, and the product 3 3 d is the fuper- 
 ficial content required. 
 
 Problem IV. 
 
 To meafure the trapezium O M N V. 
 
 F f Rule. 
 
I lO 
 
 Fig. IV. 
 
 Fig.V. 
 
 Of Geometrical Axioms^ &e. 
 
 Rule. 
 
 I. Draw the line MV^ and then the trapezium is divi¬ 
 ded into two triangles. 
 
 1. Meafurc each triangle feverally, and the fums toge¬ 
 ther, and the total will be the fuperficial content required. 
 
 Problem V. 
 
 To meajnre any irregular figure, as the figure L, O, V. 
 R, S, N, M, 1 , D, E, W. 
 
 Rule. 
 
 I. Divide the figure into triangles, and meafure each 
 triangle feverally, and note its quantity. 
 
 Add all the triangles together, and the total will be 
 the fuperficial content of the figure required. 
 
 Problem VI. 
 
 To ?neafure any regular polygon, as a pent agon,hex agon, 
 heptagon, oCfagon, nonagon, or decagon. 
 
 For the menfuration of all thole regular polygons, there 
 IS one general rule, o^iz. 
 
 Multiply half the circumference, as E, O, M, I, by 
 the radius or femidiameter AN, and the produd; will be 
 equal to the fuperficial content required. Or otherwife, 
 you may divide the figure into triangles, and then meafure 
 each triangle, and add up the fum total of the whole, and 
 that fliall be the fuperficial content required. 
 
 Problem VII. 
 
 The fide of a pentagon, &c. as BA ghen, to find the 
 femidtayneter of a circle injerihed therein. 
 
 Rule. 
 
 Fig. XV. fJ-fj fo is the fide of the polygon (be it 
 
 any whatfoever) to the radius of the circle inferibed 
 therein. 
 
 Problem VIII. 
 
 To meafure any circle, or JeBion of a circle. 
 
 The diameter of every circle hath fuch proportion to 
 its own circumference, as 7 hath to xx, or rather as 
 113 is to 3 j-y, therefore if any one be given, the other 
 may thus befound. Rule 
 
 rig. VI. 
 
Ill 
 
 Of Geometrical Axioms^ &c. 
 
 Rule I. 
 
 The diameter being given, to Jtnd the circumference. 
 Praftice. 
 
 Multiply the diameter given by and the produft 
 divide by 7, the quotient is the circumference required. 
 
 Rule II. 
 
 The circumference being given, to find the diameter. 
 Pradlice. 
 
 Multiply the circumference given by 7, and divide the 
 produd by xx, and the quotient is the diameter required. 
 
 Rule III. 
 
 The diameter of a circle being given, to fnd the area. 
 Pradlice. 
 
 I. Multiply the diameter into itfelf, and that produdl 
 multiply by II. 
 
 X. Divide the lall produdl by 14^ and the quotient is 
 the area required. 
 
 Rule IV. 
 
 The circumference of a circle being given, to find the 
 area. 
 
 Pradlice. 
 
 I. Multiply the circumference given, by itfelf, and the 
 produdl alfo multiply by 7. 
 
 X. Divide the laft produdt by 88, and the quotient will 
 be the area required. 
 
 Rule V. 
 
 The circumference and diameter of a circle being given 
 to find the area. ' 
 
 Pradlice. 
 
 Multiply half the circumference by half the dia¬ 
 meter, and the produdl will be the area required. 
 
 Rule VI. 
 
 The area of any circle being given, to find the diameter. 
 
 4 Practice. 
 
 
 
Divide the area given by 11, and the quotient is the 
 diameter required. 
 
 Or thus. 
 
 As rx, is to aSj fo is the area to the diameter required. 
 Rule. VII. 
 
 The area of any circle being given, to find its circum¬ 
 ference. 
 
 Prafbice. 
 
 As 7 is to 88„ fo is the area to the fquare of the 
 circumference, whofe root is the circumference required- 
 
 Rule VIII. 
 
 The area of any circle being given, to f ind the fide of 
 a fquare equal thereunto. 
 
 Praftice. 
 
 Extradl the fquare root of the area given, and the root 
 fliall be the lide of the fquare required. 
 
 Rule IX. 
 
 The diameter of any circle being given, to find the 
 fide of a fquare, the content of which fquare Jhall be e- 
 qual to the fuperficiat content of the circle, whofe dia¬ 
 meter was given. 
 
 Pradlice. 
 
 As 7 is to XX, fo is the fquare of the radius to the area 
 required. 
 
 Or thus. 
 
 As 113 is to gyy, fo is the fquare of the radius to the 
 area required. 
 
 Rule X. 
 
 The diameter and curve line of a femicircle being given, 
 to f ind the content. 
 
 Pradlice. 
 
 Multiply j the curve by the radius, and the produdt is 
 the content required. 
 
 Rule 
 
Of GeQm6$xical Axioms, &d 
 
 ■ Rule XL 
 
 The radius and curve line of a feclor of a circle being 
 given, to find the content. 
 
 Pradlice, 
 
 Multiply the radius by \ the curve line^ and the pro¬ 
 duct is the content required. 
 
 Rule XII. 
 
 Any part, or fegment, of a circle being given, as BC 
 D L, fig. XVIII. to find the area thereof. 
 
 Pradlice. 
 
 1. By problem XL feft. 11. part 1. find the center E of 
 fhe arch BCD, and draw the lines B E and D E, which 
 will complete the quadrant E B C D. 
 
 2 . By the laft rule, meafure the whole fedlof EBD, 
 and from it deduft the triangle EBD, and the remainder is 
 the content of the fegment required. 
 
 Problem IX. 
 
 • • ^ 
 
 To meajure any ellipfs or oval fornix 
 Rule. 
 
 Multiply the longeft diameter by the Ihorteft, and ex- 
 traa the fquare root of the produft, which fquare root 
 lliall be the diameter of a circle equal to the ellipfis, which 
 being given, you may by the preceding problem find the 
 area required. 
 
 Problem X. 
 
 To meafure the fuperfcies of a fphere, or hemi- 
 fphere. 
 
 Rule. 
 
 The fuperficies of a fphere is equal to four great qircles 
 of that fphere, therefore find the area of a circle, whofe 
 diameter is equal to the diameter of the fphere given, and 
 four times that area is the area of the fphere required, 
 and confequently the half is the area of the hemifphere 
 alfo. 
 
1 14- 
 
 Of Geometrical j4xioms, &c. 
 
 Or thus, 
 
 Multiply the diameter by the circumference, and the 
 produdl lhall be the fuperficial content of the Iphere or 
 giobe. And if the axis only is given, the fuperficial con¬ 
 tent mav thus be found, viz. as 7 is to xx, fo is the fquare 
 of the diameter to the content required. 
 
 Problem XL 
 
 To 7neafure the JuperJicial content of any conC. 
 
 Practice. 
 
 I. Find the fuperficial content of the bafe by problem 
 VIII. hereof 
 
 1; Multiply the length contained between the vertex 
 and the circumference of the bafe, by the circumference 
 of the bafe, and to the product add the content of the 
 bafe, and the total is the content required. 
 
 Problem XII. 
 
 To meafure tlie fnperfkml content of any pyramis. 
 
 I. By the definition 33. fedl. I. part I. a pyramis is 
 comprehended under divers flat fuperficies, whofe areas 
 being found by the preceding problems and added to¬ 
 gether, their total will be the content required. 
 
 Prop. I.EM XIII. 
 
 To find the fuperficial content of a cylinder. 
 
 Practice. 
 
 I. Find the area of both ends by problem VIII. hereof, 
 as alfo its circumference, which multiplied into the length, 
 and the produdl added to the areas of both ends, their to¬ 
 tal is the content required. 
 
 Problem XlV. 
 
 T» meafure the fuperficial content of any fragment or 
 part of a glohe, or Jpbere. 
 
 Pradtice. 
 
 As tlie whole diameter of the globe is to the fuperfi¬ 
 cial content of the globe, fo is that part of the diameter 
 belonging to the part, or fragment of the globe, to the 
 fuperficial content required. Sect. 
 
11 $ 
 
 Plate XVII.' 
 
 Of Geometricul Axioms forthe Menfuration oj 
 
 folid Bodies:^ 
 
 ;j|)I h: 
 
 ; njiilfli 
 
 Problem 1. 
 
 1 
 
 To Tueafure the folidity of a cube, (ts the cube ABCD, 
 Tnhoje fide is equal to 3 foot. ^ 
 
 Proportion. ^ 
 
 As I is to 3 the breadth, fo is 3 to 9, ,and 9 to 1.^, 
 the folidity required. ' ; 
 
 Or thus, y 
 
 .If • . i 
 
 I. Multiply the fide 3 by 3, the produdl is 9. 
 
 a. Multiply the laft produdl 9 hyi 3 the depth, and the 
 
 produft will be xy, the folidity required. 
 
 ) 
 
 Kl. ’Tis to be obferved that a parallelopipedoii is but 
 a long cube, and that the above rules, or proportions 
 will meafure the lame, therefore an example is peed- 
 lefs. See fig. X. 
 
 P R 0°B L R M II. 
 
 To meafure a pyrands, as the pyramis M A 0 N, 
 whofe baje is a geometrical fquare, having each of its 
 fiides equal to 6 , and its altitude to xz foot. 
 
 Rule. 
 
 I. Find the area of the bafe. 
 
 Fig. XI. 
 
 z. Multiply the area by of the altitude, and the 
 produ6l will be the folidity required. And as this is the 
 rule by which a cone is alfo meafured (as the cone, fig. 
 XII.) therefore ’tis needlefs to add an example. 
 
 Problem 
 
^ To meafure the frujlum nf a cone, or pyramis, as the 
 JrnJhm A and B, m figures XIII, and XIV. 
 
 1. Find the area of each end of the fruftuin. 
 
 1. Multiply one area by the other, and extraft the 
 fquare root of their produd. 
 
 5- Add this fquare root to the fum of both areas, and 
 then- fum multiply by ^ of the fruftum’s lenqth, and 
 that produd lliall be the true folidity required. 
 
 ^ If you find the areas in inches, you mull divide the 
 folidity fo found, by 1718, the ntimber of cubical 
 inches m a cubical foot, and the quotient will be 
 the content in feet. 
 
 I do advife the young fludent to be perfed in this 
 problem, for hereon depends the whole truth of timber 
 nieafuiing, which hitherto has been kept in the dark to 
 the great injury of all gentlemen, who have difpofed of great 
 quantities, of timber according to the cuftomary (tho’ falfe 
 and bale) way of meafuring. 
 
 Problem IV. 
 
 To meafure the folidity of a fphere, glohe, or hall. 
 
 Suppofe the diameter of a fphere, globe, &c. be ix in¬ 
 ches, what is the folidity thereof.? 
 
 » 
 
 Proportion. 
 
 As II IS to II, fo is the cube of the diameter to the 
 folidity required. 
 
 Or thus. 
 
 Cube the diameter, multiply by ii, and divide by ai. 
 pee the operation. 
 
 Diameter 
 
Of Geometrical Axioms^ &c. 
 
 Diameter ix 
 
 iz 
 
 Produdl 144, 
 
 IX 
 
 x^ 
 
 17x8 The diameter cubed. 
 Multiply by n 
 
 17x8 
 
 17x8 
 
 Divide by 11^19008(907, folidity required. 
 / i 89-- 
 
 108 
 
 107 
 
 i remains. 
 
 Problem V. 
 
 folidity of a Jphere being given, to find its dia¬ 
 meter or axis. 
 
 Pradlice. 
 
 in 
 
 As XX is to 4x, 
 required. 
 
 fo is the folidity to the axis or diameter 
 Problem VI. 
 
 ^ fe&ne?it or portion of a fphere being given, to find 
 its axis. 
 
 Pradlice. 
 
 I. Multiply i- the chord of the fegment, and divide the 
 produdl by the height of the fegment. 
 
 X. For the quotient add the height of the given por¬ 
 tion, and the fum is the axis required. 
 
 Problem VII. 
 
 To meafure the folidity of a cylinder. 
 
 Rule 
 
 I. Find the fuperficial content of one end. 
 
 X. Multiply the area fo found by the length, and the 
 produdl is the content required. 
 
 H h 
 
 Problem 
 
ii8 
 
 Of Geometrical Axioms, &c. 
 
 Problem VIII. 
 
 To meafure the folidity of a prifm. 
 
 I. Find the fuperficial content of one end. 
 
 X. Multiply the content fo found, by the length, and 
 the produft is the folidity required. 
 
 Problem IX. 
 
 To meafure the folidity of a mount, terrace walk, ca¬ 
 nal, &c. 
 
 Let A B C D E F reprefent the profile of a mount, ter¬ 
 race walk, &c. and tis required to meafure the folidity 
 thereof 
 
 Rule. 
 
 I. The hopes ABE and CDF, are prifms, there¬ 
 fore meafure thofe parts according to the laft problem. 
 
 a. The body, or midft B C E F, is a long cube. There¬ 
 fore meafure that part according to problem I. hereof 
 
 g. Add both their quantities together, and the total 
 fliall be the folidity required. 
 
 8^ This kind of meafure is always meafured by the cu¬ 
 bical yard (which by gardeners is called a load) and 
 contains ay cubical feet. Therefore if the dimenfions 
 be taken in feet, the folidity muff be divided by ay^ 
 and the quotient will be the folidity in yards. 
 
 The profile fig. XVII. is a reprefentation of the infide 
 of a canal, fifli-pond, &c. and is meafured by the very 
 fame rule as the above. Therefore to repeat the fame again 
 is needlefs. 
 
 Thefe three laft fections being duly conlider’d, and well 
 underftood (which may foon be done) the young ftudent 
 will thereby be enabled to meafure tlie fuperficial or folid 
 content of any figure or body whatfoever. And feeing 
 that the moft difficult part of the menfuration of building, 
 land, &c. confifts in the manner of taking the dimenfions, 
 therefore that lhall be the work of the next fedlion, to 
 which I proceed. 
 
 Sect. 
 

 Sect. IV. 
 
 Of tie federal Meafures that Artificers work is ac¬ 
 counted by^ and the Manner of taking their Di~ 
 menfions. 
 
 I. Of Carpenters wori. 
 
 I.' I ’ H E principal work of carpenters is flooring, par- 
 ■*• titioning, and roofing, all which are meafured by 
 the fquare, or loo feet produced by lo feet fquared or 
 multiplied into itfelf 
 
 а. When you are to take the dimenfions of the frame 
 of any timber floor, you mull allow for the length of the 
 join laid in the walls, which is generally 9 or i o inches. 
 
 3. When you meafure flooring, without joift, the di- 
 menlions are to be taken to the extreams thereof out of 
 which you mull; deduct the well-holes of the ftair-cafe, and 
 the chimney ways. 
 
 4. When you meafure partitioning, you mufl: dedufl: 
 doors and door-cafes, and windows alio, provided they arc 
 not to be included. 
 
 S- When you meafure roofs, meafure the length of the 
 rafters by the length of the roof, and afterwards the hypps 
 lingly (inftead of the common way, by allowing one 
 and the fuperficial content of the ground plot) without 
 making any deductions for the holes of the chimney lhafts, 
 or vacancies, for sky or lanthorn lights, except 'tis agreed 
 on otherwife. 
 
 б . Doors, fliop-windows, &c. are meafured by the fquare 
 foot, and alfo fafli frames, &c. ftairs and flair-cafes are 
 accounted for by the Ilep, in proportion to the nature and 
 goodnefs of the work. 
 
 7. There be divers forts of work meafured by running 
 meafure, viz. in length only ; fuch as cornices in general, 
 pent-houfes, timber fronts, rails and balafters, guttering, 
 lintelling, skirting boards, brellfomers, benching, fliel- 
 ving, &c. 
 
 II. Of Glaziers work. 
 
 Glaziers work is meafured by the fquare foot, and the 
 dimenfions are taken in feet, inches, and parts, and the 
 4 moll 
 
Of the feveral Meafures, 6cc. 
 
 inoft material things to be obferved therein, are the fol¬ 
 lowing. 
 
 I. That in meafiiring of glazing in one building, there 
 are many times windows of one magnitude, and at fuch 
 times you need meallire but one, and thereby account for 
 the others. 
 
 a. That fcmicirclc, ovalar, &c. windows be meafured, 
 as fquare windows, whofe breadths, &c. are equal to their 
 diameters ; and the rcalbn for fo doing is, bccaufe there is 
 great wafte in cutting the glafs, and much more time ex¬ 
 pended therein, than if the whole was a fquare window. 
 
 III. Of Joiners work. 
 
 1. Joiners work is meafured by the Iquare yard (of 9 
 feet) but their dimenfions are taken in feet and inches, 
 and the product; being divided by 9 (the fquare feet in a 
 yard) the quotient is yards. 
 
 2. In taking tire dimenfions you mult obferve the fol¬ 
 lowing rules. 
 
 1. In taking the Ireight of any corniflr, wainfeot, &c. 
 tliat \’ou meallire with a line into, and about, every moul¬ 
 ding, as is contained between the cicling and the floor, 
 wliich you mult call the height, or breadth, and the cir¬ 
 cumference of the room meafured on the floor, the length, 
 w hich being multiply'd as is taught in feft. I. hereof, and 
 divided by 9, the quotient is the content thereof 
 
 2. When you meafure w'indow-fliutters, doors, draw¬ 
 ers, feats, or pews in clrurches, &c. you mult account the 
 meafure (as much more that it contains, in regard to its be¬ 
 ing worked on both lides, and is what w'orkmen call w'ork 
 and ( work. 
 
 5. That in meafiiring wainfeot you ahvays dedufl the 
 doors, chimneys, and window's contained therein. 
 
 4. That you meafure the window boards, fipdietas, 
 cheeks, skirt boards, &c. by themfelvcs. And, 
 
 Lajlly, When joiners make cornice and bafe, and fub- 
 bafe, &c. fingly, they are meafured by running meafure, 
 as alio architrave and freizc. 
 
 N. B. That chimney-pieces, frontifpieces of doors, or¬ 
 naments of w'lndows, pediments, &c. arc vidued ac¬ 
 cording to their goodnefs, at - - - per piece. 
 
 IV. Of Painters w'ork. 
 
 I. Painters work is meafured and taken as joiners, both 
 in refpecl to girting about the moulding, as well as in 
 
 meafiiring 
 
Of feveral Meafures, 6cc, 
 
 meafuring the length on the circumference of the floor, 
 &c. and the dednidions to be made are the fame, but in- 
 ftead of accounting doors, window-fluitters, &c. work and 
 half work, they account it all whole work. 
 
 1. Window-lights, bars, cafements, &c. are done at- 
 
 per piece, and oftentimes cantalivers, modillions, &c. and 
 ornaments between them. 
 
 V. Of Plalterers work. 
 
 Plaftcrers works are principally of two kinds, viz. cieling 
 work which is lathed and plaftered, and rendering, which 
 is alfo of two kinds, viz. rendering upon brickwalls free 
 from quarters, &c. and rendering hi partitions between 
 quarters, which are all meafured by yard meafure, taken 
 by feet and inches, and reduced into yards, as before de¬ 
 livered. 
 
 The principal things to be obferved in taking the di- 
 menlions are the following. 
 
 I. To dedudl chimneys, windows, and doors. 
 
 a. To make no deduflions (in rendering upon brick) for 
 doors or windows, by realbn tlie jaums and heads, general¬ 
 ly exceed the dimenlions of the vacancies. 
 
 3. That fuch fommers and girders as lie below a cieling 
 be dedufted, if the workman find materials, otherwife not. 
 
 4. In rendering, when materials are found by the work¬ 
 man, to dedubl ~ for the quarters, but when workmanfliip 
 only is found, no deduftion mull be made, for the work¬ 
 man could have rendered the whole as foon as if there had 
 been no quarters there. 
 
 ). When you meafure whiting and colouring between 
 quarters, you muft add a fourth or fifth part, for the returns 
 or fides of the quarters. 
 
 Laftly, Ornaments in plafter, as ornaments in cielings, 
 capitals, architraves, freizes, cornices, &c. are meafured 
 
 by foot meafure, in length only at- per foot according 
 
 to the goodnefs and nature of the work. 
 
 VI. Of Mafons work. 
 
 Mafons work is meafured three different ways, as firfl, 
 running meafure, as the coping of walls, &c. Secondly, 
 fuperficial, as pavements, &c. And laflly folid, as blocks 
 of marble, &c. which feveral meafures being all perfor¬ 
 med by the 3 firfl fedlions hereof, I need fay no more there¬ 
 of, but that their dimenlions are taken in feet, inches and 
 parts. 
 
 1 i VII. Of 
 
 I2E 
 
 
 
 
 :.W3^\'Wjr 
 
22 
 
 Of feveral Meafures, &c. 
 
 VII. Of Bricklayers work. 
 
 Of bricklayers work there are divers kinds, but the prin¬ 
 cipal are walling, tyling, and paving. 
 
 I. Of walling, performed by the rod. 
 
 I. Of walls there are divers kinds, in refpedl to their 
 length, height and thickncfs’s, whofe dimenfions are always 
 taken in feet and inch ineafure in refpedl to length and 
 height, and by the length of a brick, &c. in refpedl to 
 their thicknefs. 
 
 а. The meafure by which brickwalls are accounted is 
 a fqtiare rod or id feet 6 inches fquarcd, whofe produdl 
 or quantity, is xyi feet fquare and ^6 inches, or >1^, whofe 
 r is feet and quarter d8 feet 
 
 5. The manner of meafuring brick walls is the very fame 
 as any other fuperficial meafure, provided their thicknefs 
 be exadlly the ftandard thicknefs, viz. one brick and ~ and 
 the produbl; of tlie dimenfions divided by xd, where¬ 
 by the number of fquare rods contained therein may be 
 known. 
 
 4. When the thicknefs of brickwalls exceeds or is lefs 
 than the flandard thicknefs of one brick and half, they mull 
 be reduced thereunto by this general rule. 
 
 Multiply the fuperficial content of the wall by the num¬ 
 ber of half bricks contained in the thicknefs, and divide 
 the produtl by three (the number of bricks contained in 
 the ftandard thicknefs of one brick and ~ ) and the quo¬ 
 tient fliall be the true content of the wall, reduced to the 
 ftandard thicknefs of one brick and half, as required. 
 
 S’. When brickwalls are of divers thicknefs they muft 
 be feveraly taken, and their feveral quantities being added 
 together will be the content of the whole, as required. And 
 here note, that whatfoever doors, windows, &c. are con¬ 
 tained in the feveral thicknelles of fuch walls, that you 
 dedufl them out of the total product of the refpedlive di- 
 menlions or thicknefs, wherein they are fttuate, and the 
 remainder will be the true content of the work. 
 
 б . When you are to meafure walls that meet, and confti- 
 tutc an angle, you muft take the length of one wall to 
 the out-fide of the angle, and the other to the inlide. 
 
 7. When you have any chimneys to meafure, meafure 
 them as a folid, and deduQ: the vacancies, (as taught by 
 Venteriis Mandey in his appendix of chimneys reformed 
 in his Mellificiiim Menfionis) and thereby you will have 
 
 the 
 
I23 
 
 Of feveral Meafures^ &c. 
 
 tlic true folidity ; but if you pradtice tlie common ay of 
 girting chimneys^ you never can have the true content, 
 and will always remain in the dark, as many llubborn ig¬ 
 norant conceited fools now are. 
 
 r. Of walling, performed by foot meafure. 
 
 I. This part of walling is that which is called ornament, 
 fuch as arches over doors, windows, &c. Facias, archi¬ 
 traves of doors, windows, &c. freizes, cornices, ruftick 
 cones, rubbed returns, &c. and in lliort all kind of work 
 performed in a rubbing houfe with ax and ftone is orna¬ 
 mental work, and is always performed at- per foot. 
 
 a. When you have any ol' thefe ornaments to meafure, 
 that have unequal lides, as the arch over a window, &c. you 
 muft take the dimenfion thereof in the middle and there¬ 
 by ’twill be a mean. And befides the aforefiid ornaments 
 which arc performed by foot meafure, there are divers o- 
 ther ornaments that are performed -piece, and fuch are 
 peers, columns, pillafters, architraves, freizes, cornices, 
 grottos, cafeades, pediments, &c. which are valued accor¬ 
 ding to the nature and goodnefs of the materials and work- 
 manlliip. 
 
 3. Of tyling. 
 
 I. As carpenters meafure their roofs by the fquare of 10 
 feet {viz. 1 00 ) fo alfo do bricklayers their tyling, whofe 
 dimenfions are always taken in feet and inch meallire, and 
 their produdts being divided by 100 (the number of fquare 
 feet in a fquare) the quotient is the content required. 
 
 t. When you take the dimenfion of a roof you muft 
 firft meafure the whole length, as far as the tiles are laid, 
 for your length, and trom the ridge to the eves for the 
 depth, and thereby the quantity of tyling will exceed the 
 quantity of rooftng, by fo much as the tyles go beyond 
 the roof at each end, and over the eves board. 
 
 3. It often happens, that in fome roofs there are many 
 hips and valleys, which muft be paid for, at -—per foot 
 running meafure. 
 
 N. B. That what is here faid of tyling, the fame is be 
 underftood of flating. 
 
 4. Of paving. 
 
 Since it often happens that cellars, kitchins, grottos, 
 &c. are paved by bricklayers, therefore I thought it necef- 
 fary to mention it here, at the conclufion hereof where- 
 
 I in 
 
i 24 
 
 Of the Manner of cajiing up 
 
 in you are to underftand, that the dinienfions of fuch 
 work are taken in feet and inch meafure^ and the content 
 is always given in fquare yard meafure^ as plaftering, ren¬ 
 derings &c. 
 
 
 Sect. V. 
 
 Of the Manner of cafling up the Dimenfons of Land 
 Meafure taken with Gunter’t Chain (which of 
 all others is the hejiS) 
 
 Problem I. 
 
 Suppofe an oblong piece of land contain ly chains xj- 
 linlis in length, and 13 chains 75- links in breadth, ivhat 
 is the content ? 
 
 Rule. 
 
 I. Multiply 15- : xy, by 13 : 7T, according to the com¬ 
 mon way of vulgar multiplication, and the produft will 
 be xo,96875-, from which cut off the five lall figures to¬ 
 wards the right hand, viz. 96875-, and the remainder to 
 the left, -viz. xo, is the number of acres. 
 
 X. Multiply the five figures cut off by 4 (the number 
 of roods in an acre) and the produtl will be 3875-00, 
 from which cut off' five figures, as before, and the remain¬ 
 ing is roods. 
 
 3. Multiply thole figures laft cut off by 40 (the num¬ 
 ber of fquare poles in a rood) and the produdt is 35-00000, 
 from which cut off five figures, as before, and the re¬ 
 mainder is poles. 
 
 Lajlly, If any numbers remain in the laft five figures 
 cut oft', multiply them by xyx^, and cut off five figures, 
 as before, and the remainders to the left, ftrall be the odd 
 feet, which in land meafure is exadl enough. See the 
 operation. 
 
 Length 
 
125 
 
 the Dimenjions of Land Meafure, See. 
 
 Length iS-'i-T 
 
 Breadth 
 
 'l6rs 
 
 10675- 
 
 ^sns 
 
 15-15- 
 
 Acres 10)96875- Thefe five figures 
 Multiply by 4 The roods in an acre. 
 
 Rods 3)875-00 Thefe five figures 
 
 Multiply by 40 The poles in a rod. 
 
 Poles 3 5-)ooooo 
 
 Problem II. 
 
 Plate XVII. 
 
 The plan of a piece of land 'with the area given, to 
 find the fcale hj lohich it was plotted, fuppofmg fuch a 
 fcale was left. 
 
 Suppofe A B, C Dj to be a plan, equal in area to 34 
 acres 31 centefims, I demand by what fcale the figure was 
 plan’d. 
 
 I. If you meafure the fide AB with a fcale of 10 in 
 an inch, the length A B will contain 38 chains and 11 
 centefims, and the breadth A C 6 chains and 15- cente- rig. vit 
 fims. The content will be found to be xg acres and 81 
 parts. Wherefore if you divide the diftance on the fcale 
 of logarithms between 13 : 81, and 34:31 into two equal 
 parts, and fetting one foot of your compalfes upon 10, the 
 imagin’d fcale, the other will reach to 11, which is the 
 fcale required. 
 
 Problem III. 
 
 Of the menf ration of turf, with which grafs walks, 
 plotts, &c. are tnade. The tarfufed in thefe works, is the 
 finefi that can he had, from commons, heaths, &c. which 
 are generally cut at one fnlling per 100, every turf be¬ 
 ing one foot in breadth and three foot in length. There¬ 
 fore to find what quantity of turf will cover any walk, 
 
 &c. find how many fquare feet are contain d therein, and 
 divide that number by 3, the number of feet in a turf, 
 and the quotient will be the number of turf required. As 
 for example, 
 
 K I 
 
 4 
 
 There 
 
1 26 
 
 Of the Manner oj cajiing up 
 
 There is a walk to be turf’d, whofe length is loo foot, 
 and breadth i6 feet, how many three foot turt will cover 
 the fame, fuppoling no waltc to be made ? 
 
 The length loo 
 
 The breadth l 6 
 
 IToduCl I boo Which divide by 5, as follows. 
 
 3 Tiboo/37 the number of tiui re- 
 
 (quired. 
 
 lo 
 
 lo 
 
 _9 
 
 I remains. 
 
 Note, That this calculation fuppofcs no waftc to be made, 
 wliich in laying them is impollible. Therefore the ufual 
 allowance for walte is as follows, mz. a hundred of turf, 
 nhich contains 300 loot, is allowed to completely fi- 
 nilh one rod of ground, which contains ^7^7 feet. 
 
 Sect. VI. 
 
 Of diuers /InulogieSy or Proportions, in Land Mecfure. 
 Proportion I. 
 
 Having the length and breadth of an oh long given in 
 chains, to find the contents in acres. 
 
 As 10 is to the breadth in chains, fo is the length in 
 chains to the content in acres. 
 
 Proportion II. 
 
 Having the perpendicular and hafe of a triangle given 
 in perches, to f irid the content in acres. 
 
 As 310 is to the perpendicular, fo ,is the bale to the 
 content in acres. 
 
 Proportion 
 
I 
 
 y'/.^^.■xvx 
 
 Jt> 6 
 
 
 
 39S .36 
 
 Flg-:X. 
 
 ^<yUddi/ Jf 00:00 
 
the Dimenftons of Land Meafure, See. 
 
 Proportion III. 
 
 Having the perpendicular and hafe of a tria 7 tgle givefi 
 in chaim, to find the content in acres. 
 
 As to is to the perpendicular, fo is the bafe to the 
 content in acres. 
 
 Proportion IV. 
 
 Having the content of a fiiperficies in one hind of med- 
 fare, to find the content of the fame fuperficies, accor¬ 
 ding to any hind of perch meafure. 
 
 As tlic lengtii of the fecond perch is to the length of 
 the firlt perch, fo is the content in acres to a fourth num¬ 
 ber, and the fourth number to the content in acres re¬ 
 quired. 
 
 Proportion V. 
 
 Having the length and breadth of an oblong Juperfides 
 given in perches, to find the content in acres. 
 
 As I do is to the breadth in chains, fo is the length in 
 perches to the content in acres. 
 
 Proportion VI. 
 
 Having the length of a fiiperf icies in chains, to find 
 the breadth of an acre. 
 
 As tile length in chains is to ten, fo is one acre to the 
 breadth in chain meafure. 
 
•i8 
 
 THE 
 
 PRACTICE 
 
 O F 
 
 Architcdiire, Gardening, Menfiiration, and 
 
 Land-Surveying, Geometrically demonftrated. 
 
 PART IV. 
 
 Containing divers excellent Tables of Menfurati- 
 on, which fhew^ by infpeclion, the true fu- 
 perficial, or folid Content, of any Kind of 
 Meafure, according to any Dimenfions given. 
 
 Sect. I. 
 
 Of Englilh Meafures ufed in Lands and Buildings. 
 
 "D EFORE I begin the tables, ’twill not be improper to 
 ' inl'ert the following meafures, viz. That 
 
 A fquare foot 
 A cubical foot 
 A fquare yard 
 A cubical yard 
 
 144 fquare inches. 
 17x8 cubical inches. 
 9 fquare feet. 
 
 117 cubical feet. 
 
 A 
 
Of Englilli MeafureSy See. 
 
 A fquare 
 
 A load of timber 
 
 r 
 
 Z 
 
 A load of thick, 
 
 planks 
 
 A geometrical pace 
 A geometrical perch 
 A Itatute pole or perch 
 A fquare ftatute perch 
 A woodland pole or perch 
 A fquare woodland pole 
 A forreft pole or peixh 
 A fquare forreft pole 
 4 ftatute perches 
 10 chains length 
 4 chains length 
 40 fquare perches 
 4 rood, or 160 perches 
 A hide of land 
 
 100 fquare feet, or 10 foot 
 every way. 
 yo foot cubical. 
 
 300 fquare feet, 
 loo fquare feet. 
 
 I yo fquare feet. 
 
 400 fquare feet, 
 doo fquare feet, 
 y feet ^ 
 
 10 feet (in length, 
 id feet 
 
 x7at fquare feet. 
 
 18 foot in length. 
 
 3 a4 fquare feet. 
 
 II foot in length. 
 
 441 fquare feet. 
 
 One chain length, 
 
 A furlong or acre’s length. 
 An acre’s breadth. 
 
 A rood or ^ acre. 
 
 An acre. 
 
 100 acres 
 
 I 29 
 
 Bricks according to the ftatute, foould be 9 inches in 
 length, 4 inches ^ in breadth, and x inches ^ in thicknels; 
 500 is a load. Plain tile, in length 10 inches breadth 
 6 inches f, and thicknefs J-inch; 1000 is a load. Gutter 
 tile in length 10 inches the breadth and thicknefs in 
 proportion. Roof tile in length 13 inches, thicknefs ^ 
 inch and r quarter, the depth proportional. Lath y fcore 
 to the bundle, when y foot long; but when 4 foot in 
 length, then d fcore to the bundle. Deals and nails i%o 
 to the hundred. Lime is fold by the bag, which fliould 
 be a bulhel, ay bags is called a hundred; ’tis in fome 
 places fold by the load, which is about 40 bufliels. A 
 tun of iron is aa4o pound weight; and a fodder of 
 lead 19 hundred r, or xi84 pound. 
 
 L 1 
 
 Sect, 
 
Of the Explication of the Infpeclional Tables of 
 Menfuration. 
 
 Table I. 
 
 '^His table is defign’d for fmall menfurations, as paint- 
 ing laid with leaf gold, which is generally done for y j. 
 per foot, if plain without carving; and if carved, double 
 the price, by reafon a great quantity of gold is wafted in 
 gilding the broken parts of the ftime and glazing, &c. 
 which is coftly work, and feldom is in great quantity to¬ 
 gether. 
 
 The life of this table is as follows. 
 
 Suppofe a piece of gilding, glazing marble, &c. be 11 
 inches in breadth, ivhat length mujt he taken for a fquare 
 foot ? 
 
 Pradlice. 
 
 I. In the firft column (entitled the dimenfions breadth 
 in inches) find ii (the breadth of the work) and againft 
 it ftands i, i, i, which lignifies one foot, one inch, and 
 one tenth part ot an inch, and is the length required to 
 make one fquare foot. 
 
 a. To find the content of the whole, open a pair of com- 
 pafles to the extent of i foot, i inch, and f„, and run that 
 extent through the whole length of the dimenfions, and the 
 number of thofe extents fliall be the number of feet requi¬ 
 red. 
 
 Example 2-. 
 
 Suppofe a Jlabe of marble is 13 foot 4 inches and of 
 an inch in length, and xg inches wide at one end, and 17 
 at the other, what length mufi be taken for a fquare 
 foot, and how many doth it contain ? 
 
 I. Add the ends together zg and 17, and take a mean, 
 wz. zo. 
 
 z. A- 
 
Of the Explication, <Scc. 
 
 2. Agaiiifl; to in the firft column Hands o, x, %hz. 
 feven inches and r. of an inch, which is the length of a 
 foot required. 
 
 And if the coinpalfes be opened to that extent, ’twill pafs 
 through the lame exaflly xx times, which is the number 
 of feet contain'd therein. 
 
 Kote, That what is faid here in relation to the menfu- 
 ration ot marble, the fame is to be underllood in any 
 other kind of meafure. 
 
 Table II. 
 
 This table is calculated for any large menfurations of 
 fuperficial feet meafure, and is divided into xi columns. 
 The firft contains the dimenlions breadth in inches from 
 
 I to gd. The other xo columns, contains the dimenlions 
 length in feet. Every of thefe columns is number’d at their 
 heads from i to xo feet, as i foot, x feet, &c. and un¬ 
 der every ot thefe numbers in each column, is placed the 
 letters F P, which denote feet and part of feet. And 
 here you are to obferve, that every fquare foot is divided 
 into too equal parts, and thofe parts, or numbers written 
 under the letter P in every column, are fo many parts of 
 a hundred or loot. Therefore xy of thole parts is a quar¬ 
 ter, yo a half, and yy three quarters of a foot, and the like 
 of any other centefimal part. 
 
 Ufe. 
 
 Suppoje a Jlale of marMe he tt inches m breadth and 
 
 II foot in length, luhaf s the content ? 
 
 In the angle of meeting of xx (accounted from the fide 
 or firft column,) and 11 (accounted from the head) is 
 xo: id, which is xo fquare foot and the content required. 
 
 Example x. 
 
 There is a marble pavement ix foot in breadth and 
 17 foot in length. 
 
 io» When the breadth happens to be greater than 3d in¬ 
 ches, as herein, you mult firft work, fuppofing the 
 breadth was 3 d inches only, and note that content. 
 
 x. As often as you can find 3 d in the breadth, fo many 
 times add the firft content (as in this example is 4 times) 
 and if any part of the breadth remain, proceed as at fir ft, 
 and add that to the fum of the federal additions aiiu the 
 fum fhall be the content required. I need not add the 
 operation, by reafon 'tis fo very eafy and plain. 
 
 4 When 
 
Of the Explication of 
 
 When meafures happen unequal at each end, add theni 
 together and take a mean, as in table I. 
 
 Tables III, and IV. 
 
 Thefe two tables are both calculated for the menfura- 
 tion of folids, as timber. Hone, &c. as arc truly fquare at 
 their ends. 
 
 Ufe of table 1 . 
 
 There is a piece of fquare timber, whofe /ides are each 
 equal to inches, what length muji he taken for one 
 [olid foot 
 
 Pradlice. 
 
 Againft 13 in the firft column. Hands o, 10, x, mz. 
 10 inches and x tenths of an inch, which is the length 
 required. 
 
 Table II. 
 
 Suppofe a piece of thnher be 16 inches fquare at each 
 end, and foot in length, what is the content^ 
 
 Note, That as this table is calculated but to ten foot in 
 length, therefore find the quantity of 10 foot in length 
 firlt, and then treble it, and afterwards find the quan¬ 
 tity of 7 foot in length, and add that to the former, 
 and the fum fhall be the folk! content required. 
 
 Practice. 
 
 I. Againft 16 in the firft column, under 10 feet at the 
 head. Hands 17 • 78, which ia l y folid foot, and 78 parts 
 of a hundred. 
 
 X. Ten being contain’d 3 times in 37, therefore treble 
 17 : 78, and the fum will be 73 : 34.. 
 
 3. For the quantity of the odd 7 foot, look under 7 
 foot at the head, againft 16 of the fide, and in that angle 
 of meeting Hands ix : 44, which added to the former 
 73 ; 34, is equal to 6s : 78, wz. 6s foot and 78 parts, 
 whicli is more than 3 quarters of a foot, and is the 
 folid content required. 
 
 Note, That what is faid in this example, the fame is to 
 be underftood in all others of this nature. And be- 
 caufe ’tis feldom that timber, or ftone, happens ex- 
 adlly fquare at their ends; therefore I have here fub- 
 join’d a table of mean proportionals, by the help of 
 which any unequal fided timber may be meafured by 
 this table, as in the above example. 
 
 Table 
 
the Infpe&ional Tables of Menfuration. 
 
 Table V. 
 
 This table of mean proportionals, is calculated on pur- 
 pofe to reduce unequal lided timber, &c. to exad fquarc 
 meafure, and thereby the laft table is made capable to 
 meafure any kind of four fided timber whatfoever. 
 
 Ufe. 
 
 Suppofe one fide of a piece of timber be ic) inches, and 
 the other 7 inches, what is the mean proportional, or 
 what is the length of the fide of a fquare equal there¬ 
 unto ? 
 
 Practice. 
 
 1. Againft 7 in the hrft column Hands 084700, and 
 againft 19 Hands 127877. 
 
 2. Add thofe 2 numbers of the fecond column into one 
 Him, and ’twill be equal to 212584. 
 
 5. Divide this laH number into 2 equal parts, then will 
 the half be 105192. 
 
 4. Look for this number (or the neareH to it) in the 
 table which is 107918, againH which Hands 12, which is 
 tlie length of the lide ol a fquare equal thereunto, as re¬ 
 quired. 
 
 The fide of the fquare being thus found, enter the laH ta¬ 
 ble therewith, with any length aflign’d, and proceed as 
 therein directed (which is very plain and familiar, and the 
 folid content will be found, as required. But to make it 
 fully plain, take this example. 
 
 Let the length be 9 foot. 
 
 Firjl, Look for 12 inches in the HrH column, and un¬ 
 der 9 foot (at the head of the table) Hands 9 : o, mz. 9 
 foot, o inches, which is the folid content required. ” 
 
 Table VI. 
 
 The abfolute reafon of the conHruClion of this table, is 
 to lliew the great error and deceit as is contain’d in the 
 cuHomary way of meafuring timber, and to prevent the 
 practice thereof for the future. 
 
 The manner of ufmg this table is exaftly the fame as 
 the HrH and third. The column, wherein the words, the 
 girt, &c. are inferted, is numbered Horn 10 to 100.' The 
 other column contains the feet, inches, and tenth parts of 
 an inch, as will make a lolid foot in length at cvei"y cir¬ 
 cumference of the firH column. 
 
 M m 
 
 Example, 
 
134- 
 
 Of the Explication of 
 
 Example. 
 
 There is a piece of round timber that in the middle is 
 XX inches girt {or circumference) and /ip foot in the length, 
 what's the folid content 1 
 
 Praftice. 
 
 I. In the firft column find px the girt given, and a- 
 gainfi It Hands o, 8, o, which is o feet, 8 inches, and 
 o parts, and is the lengtli of a folid foot, at that girt. 
 
 X. Take the diltance of 8 inches in your compafles, and 
 run thejii along the piece of timber in a right line, and 
 as ol'ten as that diltance is found therein, fo jnany folid 
 feet is contain’d in that piece of timber, which in this ex¬ 
 ample is do times, and therefore the folidity is do foot, or 
 one load and ten foot. 
 
 Now for a demonftration of the aforefaid error and de¬ 
 ceit in the cullomary way of meafuring. I’ll meafure the 
 aforclaid piece of timber, according to the common way, 
 which is to double the firing by which the girt is taken 
 4 times, or to take j of the girt for the fide of a fquare, 
 and then meafure the fame as fquare timber, as follows. 
 
 I. The aforefaid girt is yx, one fourth thereof is 13. The 
 fide of a fquare (which they fuppofe to be, or at leafi fay 
 is true, tho’ infinitely from it.) 
 
 X. The piece of timber being 40 long (and the fourth 
 table hereof being calculated but to 10 foot length) there¬ 
 fore meafure one fourth only, and quadruple it, fo fliall 
 the mult be the content required. for example. 
 
 Againft 13, rheiideof the fquare, and under 10 foot 
 at the head of the table. Hands 11 -. 74, the folid content 
 of 10 feet in length, which quadrupled is equal to 4dfeet 
 and qd hundred parts, which is almoft 47 feet. 
 
 From hence it appears, that the true content by the firft 
 (and true) way, is do feet complete, and by this way not 
 quite 47 feet. Therefore ’tis 13 feet too little, or lefs than 
 the true quantity. 
 
 Now fuppofe the aforejaid piece of timber was oak, 
 which IS ne^er j old for lefs than one foilling per foot, 
 then will the aforefaid lofs of feet he xi PnlUngs at 
 leaji. Therefore, 
 
 If in do foot there is but 47 foot accounted for, in yo 
 foot there is but 39^- foot accounted for, and in every yo 
 foot, or load of timber, there is almoft ii foot of timber 
 loft, which, as before, is worth at leafi eleven lltillings. 
 
 I Now 
 
the Infpeflional Tables of Menfuration. 
 
 Now if in one load of timber eleven diillings is loft, what 
 in a hundred ? Anfwer, S'T pound. 
 
 So that from what I have here delivered, ’tis evident, 
 that all liich gentlemen as have Ibid large quantities of 
 timber, by the common way of meafuring, have been ac¬ 
 tually cheated of of the fame. But as I have taken the 
 pains here, not only to demonftrate the fame, but alfo to 
 lay down eafy plain rules and tables, ’tis hoped that the 
 fame will put a final end to all fiich impoftors dealings. 
 
 Table VII. 
 
 This table is calculated as well for the ufe of the far¬ 
 mer, &c. to divide and lay out his corn-lands, meadows. 
 See. as lor a gentleman to meafiire and let, difpofe, pur- 
 chafe, &c. when a furveyor is not to be had eafily, &c. 
 
 This table is of 4 parts, each part being divided into 
 5 - columns, and every column into two others. That of 
 the firft, entituled, the breadth of the land, hath two 
 rows of figures, thofe to the left are poles or perches, and 
 thofe to the riglit hand are quarters or fourth parts of a 
 pole or perch, and are diftinguilhed at the head, by the 
 words perch and ~ parts. 
 
 The other columns have at their heads the words 
 I rood, aiood, 3 rood and one acre, and underneath thofe 
 words the letters P. pts. &c. the letter P in every column 
 ligmfies perch, and the letters pts. hundred parts of a 
 pole or perch. 
 
 I do advife the practitioner to have a ^ pole or pearch 
 divided into y equal parts, and one of thofe parts fub- 
 divided into ten equal parts, fo will the whole be in 
 effebt divided into yo parts, and will be anfwerable to 
 the hundred parts of a perch, as is exprefs’d in the ta¬ 
 ble. Thefe divifions are belt reprefented by broad-headed 
 
 nails, with the number of the divilion engraved 
 on the head of each nail. 
 
 The ufe of this table is as follows. 
 
 Suppofe a piece of landle 7 poles in breadth, how much 
 tn length will make one rood, two rood, or an acre^ 
 
 Prablice. 
 
 Againft 7 poles in the firft column, ftands in the lecond 
 y perch 7d parts, the length of one rood, and in the fe- 
 cond 11 perch 4.x parts the length of two roods, and in 
 
 the 
 
 ‘35 
 
 
I 36 Of the Explication of &c. 
 
 the third 17 perches, 28 parts, and in the fourth per¬ 
 ches 8 5- parts, the length of an acre required. So alfo had 
 the breadth of the land been 7 poles and i quarter, then 
 would the feveral lengths be as follows, viz. 
 
 Perch. Parts. 
 
 I rood ) 7 ST.! 
 
 T. rocdCii 4 (And the like of any other 
 5 rood kid ydk breadth. 
 
 Acre jix 8) 
 
 By the example laft mention’d it appears, that as often 
 as perches and 8 hundred parts is contain’d in the length 
 of any field, as is 7 poles and in breadth, fo many acres 
 is contain'd therein, and if at laft any length is remaining 
 as is lefs than an acre, meafure off fuch a length, as that 
 for 3 roods, x roods, or i rood, &c. as you find will be 
 contain'd therein, and thereby you may have the true quan¬ 
 tity to lefs than of an acre. And to find the true niea- 
 fure of the remaining part as is lefs than one rood, divide 
 the fpace of y perch yx parts, into 40 equal parts, and as 
 many of thofe parts as are contain’d in the remaining part, 
 is the number of odd perches, and thereby you have the 
 whole content, in acres, rood, and perches, as land is ge¬ 
 nerally meafured. 
 
 A very little pradlice will make this very plain and fa¬ 
 miliar, and therefore I recommend you to the fame, during 
 which T ffiall imploy my pen in other important parts of 
 architedlure and gardening which I lhall communicate in 
 another treatife very Ipeedily for publick benefit. 
 
 F INIS. 
 
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